Say Vanakkam Adab Welcome you all to Vedanta Telugu Jai channel Welcome to our Telugu Jai channel I am Mekiran sir How are you all? I am very good I wish you all to be very good Today we are going to do 2D geometry prerequisites In this we are going to solve the questions that we will tell you the maximum concept with tricks and shortcuts I have already told you that we will go on only one shot so without any further delay let us start If the chapter is big, for example, there is an integration, if it is a very big chapter, I will take two classes in it. The first class will be completely theoretical and the second class will be all PYQs. So, let's try to complete it in one shot.
So, since we are telling one shot, we will not tell you something quickly and you will not understand it. We will discuss more questions like how he is asking us questions and go ahead without wasting time. Okay guys, so, please like the video and Let us move into the session. So, we will talk about queries later. But, class is getting late.
Let us go to class first. So, first of all, distance between two points. He gave two points.
It is very easy for us to know the distance between two points. This is point A. This is point B. The distance between A and B points is the shortest distance. This is also the distance.
If you want to go from one place to another, you can go in many ways. You can go like this and like this. But these are not the shortest distances. These are also lines. But what is the distance we have?
The formula for the shortest distance. So, the shortest distance between two points. What do you write? Square root of x2-x1 whole square plus y2-y1 whole square. Sir, first x1 comes and then x2 comes.
First x2 comes and then x1 comes. Sir, no matter how you write, it is whole square. So, x1-x2 and x2-x1 are the same. Since it is a whole square, there is no problem. So, learn this way.
So, when you say distance between two points, you should remember this immediately. Is there any doubt? Next, Section Formula.
What is Section Formula? You are given two points. He gave you two points in AB.
Draw a straight line between these two points. If you divide this straight line by C, in the ratio of M to N, he will ask what is that point. Or, if you divide AB by two points, Voice clear? If we divide AB by C, what ratio is the ratio? We can divide AB by C and ask for the ratio in reverse.
What is the ratio? First, we have to find the ratio and find the C point. For example, if the point is, So, to remember directly, what we do is cross multiplication. You do like this. Cross multiplication.
You do like this. So, we have to remember the formula by doing like that. Write it.
Write. C is equal to. Samba. Are you writing? MX2 plus NX1.
Cross multiplication, sir. Cross multiplication. MX2, NX1.
So, MX2 plus NX1 by M plus N, comma. Again, MY2, NY1. MY2 plus NY1. by m plus n. That's it.
This is the section formula. This is what we call section formula. Sir, in this section formula, it is true that there is an internal and an external formula. Since CNA point divides this internally, I wrote it internally. This is the internal section formula.
Internal. What is external, sir? If m is to n is external, if it is external, it is not m is to n, it is m is to minus n. If we say m is to n externally or just m is to minus n, both are the same.
So instead of m is to n, what happens if we say m is to minus n? Even if n is there, minus n will come in every place. So we call that external. So for external, the same formula will come, so minus will come in between. By M minus N.
What is external? It is like this. This is point A, this is point B. Point C is not inside, it is outside.
M comes with this length and N comes with this length. If you take a ratio of both, M is to N should be written. If there is an external point C, I should not use M is to N to calculate that point.
If this form is external, it should be written like this. It should be written like this. Okay? You may get confused.
But look at the one I am writing. What is the midpoint of A to B? M is between midpoint.
That is x1 plus x2 by 2, comma y1 plus y2 by 2. I know all these. All the kids in 11th and 12th class know these. Learn Telavapathy and write separately for them. This is internal section formula, this is external section formula, and this is midpoint formula. Clear?
Right. After this, finding ratio. You said to find out the ratio, right? He gave a point and b point. He gave already two points.
He gave two points x1, y1 and x2, y2. He will give another point c. C is the point x3, y3. It divides these a, b points.
But you can find the ratio. Now we have to find the ratio in the reverse. He gave the ratio. He gave the ratio m is to n.
He didn't give the c point. We have found C point, but now he is giving C point and we have to find the ratio. When we have to find the ratio, we should not take M to N as the ratio, but take K to 1. Or we should take 1 to K. So, we don't have to take K to 1, we can take 1 to K also.
So, this is the process. What we have to do is, here is a point. For example, I will do it directly with the example. Babu gave A point 1,2.
He gave B point 3,5. Let's assume that this is 1,1. This is a point called 5,5.
So, another point on this is 3,3. And you ask, in which ratio this is divided? I gave points A and B and C points. I asked the ratio of these two points divided by C points. Then what should you take as the ratio?
You should take it as k is to 1. So, I took this as k is to 1. What happens if I take it? What did I say in the section formula? Cross multiplication, we have to do cross multiplication like this. k into x2, kx2 plus 1 into x1, 1x1. How do I write this?
k into x2 is this, right? k into y is this. So I will do it like this.
5k plus 1 by k plus 1, k plus 1 comma. Again, k into 5, 5k plus 1 into 1, 1 plus 1 by k plus 1. What is this point equal to? It is equal to 3. So, if we make this equal to 3 and this equal to 3, then I get 5k plus 1 by k plus 1 is equal to 3. So, 5k plus 1 is equal to 3k plus 3. So, here 2k is equal to 2. So, k is equal to 2. Since I took k is equal to 1, I got 2 is equal to 1. Any doubt?
Very good, super sir. Sir, I understood the finding ratio. What is harmonic conjugate?
When we say harmonic conjugate, For example, points P and Q are there and points A and B are there. For example, if points A and B are two points, For example, point P is internal divided by m to n. P is dividing AB in the ratio m is to n internally. Similarly, Q is another point. Q is dividing these ABs externally.
It is dividing externally. In what ratio is it dividing? It is dividing in the ratio m is to n. That means P is dividing AB in the ratio m is to n internally.
Q is dividing AB. M is to N but externally divides. Then, what do we call these P point and Q point?
They are called harmonic conjugates of AB. P and Q divide AB internally and externally in the same ratio. This is M is to N and this is M is to N.
If we divide in the same ratio, P is called as harmonic conjugate of Q and Q is called as harmonic conjugate of P. So, P is called as harmonic conjugate of Q and Q is called as harmonic conjugate of P. So, the plan is...
PQs are harmonic conjugates for AB, so ABs are also harmonic conjugates for PQs. That means, the internal ratio of B divides PQs by the ratio of A. For example, if B is divided by PQs from X to Y, A is divided by PQs from X to Y. So, A and B are harmonic conjugates. Since P and Q are harmonic conjugates, A and B are harmonic conjugates automatically.
So, harmonic conjugates are the same. You should remember this. Just a definition is not more important than that. What is Points of Trisection?
Points of Trisection is, the ABs with two points, this AB line segment, whichever point divides this segment in 1 is to 2 ratio, that point, or whichever point divides this segment in 2 is to 1 ratio, that point, these are called Trisecting Points. Trisect is three parts. I have divided this whole line into three equal parts. One, two. This is one part, this is another part, and this is another part.
I have divided it into three equal parts. Now, this part is called the trisecting part and this part is also called the trisecting part. What does this point do? It divides in the ratio of 1 is to 2. What does this point do? It divides in the ratio of 2 is to 1. The question that she might ask is, I am giving you a point called 1, 2, and 3, 4. If he asks you to divide these two points into trisecting points, then you have to divide them into 1 is to 2 ratio first.
Then you have to divide them into 2 is to 1 ratio. If you use the section formula and divide them into 1 is to 2 ratio, you will get one point. If you divide them into 2 is to 1 ratio, you will get another point.
Similarly, you will get two points by doing trisection. So, how many points will you get by doing trisection of a line segment? Remember that 1 is 1 in 1 is 2 in 2 is 2 in 1 ratio. It leaves out the trisection. What is collinearity?
Many people say it in many ways. But I will say it in three ways. There are three points. Number one. One is collinearity.
What is collinearity? It is collinearity to be on the same line. For example, this is A, this is B, and this is C. C is here and this is here. So, what I will say is, the first point.
Sir, you find the length of AB. Call this AB. Find the distance between A and B.
Similarly, find the distance between B and C. Similarly, find the distance between A and C. Now, you find the length of AB plus... This is BC.
So, if you add AB length plus BC length, what length will I get? AC length. Is it so or not?
If it comes like this, it is called as collinear points. Why sir? It will come to the same length for everything, right? If three points are not collinear, it will not come. This is A, this is C, B is not here, it is above here.
This is AB, this is BC. Now, if I add two points, AB length and BC length, If we add ABC and ABC, will it be more or less than AC? It will be more.
Sum of the two sides of any triangle is more than the third side. Sum of the two sides. Sum of any two sides of a triangle is always more than the third side.
It will be more than the third side. This is a triangle. If it is equal, it is not a triangle. If it is equal, it is collinear. Sir, do we need ABC and ABC?
Sir, why should A be here and B here and C here? Why should A be here and C here and B here? We can write that too.
Then, AC plus CB is equal to AB. If so, we call it three points are collinear points. This is the first case. The second case is points ABC are collinear if and only if the area of the triangle ABC is zero. This is a triangle.
Does it have any area? Yes, sir. There is some area.
Delta is possible. The area is getting smaller. What is the reason for the area to become zero? The height here is zero. The point should fall on the line.
The triangle should not form. The triangle should not form, it should fall on the line. The triangle should fall down. Then what happens?
There is no triangle and there is no area. If there is no area for the triangle, then the triangle falls on the same line. So, if three points are collinear, area of the triangle will be zero.
So, area of the triangle formed by these three points ABC is zero. Then, how to calculate area of the triangle? There are many methods for that. I will tell you that now. But, even if area of the triangle is zero, we call three points collinear.
Third case is very important. Slope of AB is equal to slope of AC. If this is A, this is B, and this is C.
ABC. AB is a slope. This is a slope.
Similarly, if both the slopes are equal, then also the lines are compulsory. Should we do AB is equal to ACA? Should we not do AB is equal to BC? We can do it. If both the slopes are equal, then it is okay.
Any two slopes should be equal. AB is ACD. A, B slope is this much. It goes with this angle.
If A, C slope is the same, then A, B, C, all three are on the same line. So, even when the slopes are equal, we get the answer. One shot, right?
All the topics in one chapter will be covered. Now, about 2D geometry, you will understand when you see this class. Whether it is completed or not.
This is collinearity. Then, area of the triangle. What is area of the triangle?
I just told you that there are many methods to find area of the triangle. I told you about ladder method. You can find it in ladder method also.
This is directly debt method. What is debt method? It is half into debt. This is modulus of debt. Sir, you gave me three points.
First point is x1, y1. Write 1. Second point is x2, y2. Write 1 here. Third point is x3, y3. Write 1 here.
What is the relation of x1 and x2? If we multiply this debt with 1 by 2, the value that comes is the area of the triangle ABC. The area of the triangle ABC is this. This is a method to find out.
You know the determinant of 3 by 3 matrix, right? You can guess. I have another method too.
It is Ladder method. I can find out by using the Ladder method. What is Ladder method? Delta is nothing but half into Ladder. We have to put a constant.
How do we put a constant? How to add ladder? First point is x1,y1 Second point is x2,y2 Third point is x3,y3 Fourth point is not given to us Again we have to end with x1,y1 We have to end with the same point we started with After that what we have to do? Very simple, we have to do diagonals First diagonal, second diagonal, third diagonal These are the principal diagonals Multiply these diagonals y is equal to half into mod less half x1,y2,x1,y2 and x2,y3 and x2,y3 plus x3,y1 Now,what are the second diagonals?
Multiply these and subtract them. So,first x2,y1 minus x2,y1 minus x3,y2 minus x1,y3 That's it. This is nothing but the debt.
This is nothing but the area of the triangle ABC. Sir, will this work for all of these? Sir, it will work if you give quadrilaterals.
What will this work for? This is just a triangle, debt of 3 by 3. If you give me quadrilaterals, sir, this will not work. But this will work.
If you give me pentagon, sir, 5 points pentagon, sir, you can find its area by doing this. Not regularly. Any pentagon, any convex pentagon, you can do it for that.
Decagon, hexagon, octagon, whatever you want, you can add it to the ladder. How to write 4 points? Write x1,y1, x2,y2, x3,y3, x4,y4 and start with the first point and write it at the end.
After x4,y4, write x1,y1. Add the first principal diagonal by multiplying it. Subtract the second principal diagonal by multiplying it. Add the modulus to be positive instead of negative. Add the second principal diagonal by multiplying it by half.
Super method. Do it in this method. You can answer any question you get.
Okay? This is the way. Sir, what if I want an equilateral triangle?
Sir, what if there is a triangle that is equilateral? What is equilateral? It is the same on both sides. It is the length of one side and it is a unit.
What is the area of this unit? It is √3x4a2 where a is the length of the side of the triangle. Sir, If there is an equilateral triangle and its side length is a, root 3 by 4 into a square is the direct area of the triangle.
This is the direct area of the triangle. Sir, he gave me the height of the triangle instead of the side length. He gave me the height.
This is the height. H, sir. What happens to the area of the triangle if we give H?
If we give H, sir? Sir, H square by root 3 is the area of the triangle. H square by root 3 is the area of the triangle. Clear? If you are given side length, it is root 3 by 4 into a square.
Or if you are given height of the triangle, it is h square by root 3. So, there are two formulas for the equilateral triangle. Remember the side length and height. Will it be possible if it is there?
Whether it is there or not, the answer will be there. But we need shortcuts, right? I am telling you for shortcuts. Remember.
This is very simple to remember. Easy to remember. Remember it. Then, area of the quadrilateral.
I just said it. Babu. x1, y1, he gave one point.
x2, y2, he gave. x3, y3, x4, y4. What does delta mean?
x2, y2, x3, y3, x4, y4. What should we do next? What are we starting with? What should we do next with x1, y1?
I will write here half into modulus of this color. I will multiply and write this. x1y2 plus x2y3 plus x3y4 plus x4y1.
Now we have to do like this. 1,2,3,4. We have to do like this. What is this? x2 minus.
We have to separate. x2y1 minus x3y2. minus x4y3 minus x1y keep it like this this is the same as pentagon this is the same as any other you can think of this as the area of the quadrilateral this is the same as the points ok sir what is centroid sir?
what is the centroid? so let's talk about centroid and median here ok? so if we talk about centroid I am talking about 2D geometry.
Listen carefully. Sir, we have ABC triangle. From any point, the midpoint on opposite side. What is this?
This is the midpoint. Why midpoint? Because BC has midpoint. BC has midpoint. So, BC and D is 1, 1. So, it divides in ratio of 1 to 1. What ratio of midpoint divides?
Sameer sir has a class today, right? Yes, but sir has a throat problem, that's why he didn't take it. So, if you have seen it recently, sir is having a throat problem in the previous class.
When the top 10 topics were discussed, one of the pullers did it. So, since then, sir has a throat problem, that's why he didn't take it yesterday. But, he said he will take it today.
Let us see. So, now, D is the midpoint. So, now, if we join the midpoint on the opposite side of the ANA point, the line that comes, this line is called the median.
What is this? This is the midpoint of A, B and C. If we combine B and E, we get the median.
If we combine C and F, we get the median. The length of the median is also a formula. If we combine these three, we get the concurrent. Concurrent means that all three points go from the same point. We call that point as centroid.
We call that point as centroid. Sir, what is the formula for centroid? Sir, if the points given by you are x1, y1, x2, y2, x3, y3, then the centroid formula is average. What is this? Mean of the abscissa, mean of the ordinates.
So, this is centroid, g. x1 plus x2 plus x3 by 3, y1 plus y2 plus y3 by 3. Sir, what else is there? Centroid will divide in the ratio 2 is to 1. Centroid is this median, right?
This line is A and this is D. This line G divides the line A into 2 is to 1 ratio. Centroid divides the line A into 2 is to 1 ratio.
This line is also the same. If this line is 2, this line is 1. If this line is 2, this line is 1. So, Centroid divides the line A into 2 is to 1 ratio. There are many more properties in this line. The great thing about Centroid is that, if I take the triangle of A, G, B and consider its area, the same area will be given to A, G, C. A, G, C is also a triangle, right?
This is the area. A, G, B is one area. And B, G, C is this area. These three areas are the same.
They are equal. Do you know what is another great thing? If I join these three midpoints D, E, F, If I join these three and form a triangle, what triangle is this? Triangle DEF. What is this triangle?
Triangle DEF. There is a centroid for triangle DEF, right? So, a centroid for triangle DEF is also G. A centroid for triangle DEF is also G. Do you know another great thing?
Another thing. Here, AEF is a triangle, FDB is a triangle, D, E, C, one triangle. D, E, F, another triangle.
Four triangles are formed. These four triangles have the same area. Okay, let's talk about that now.
Okay, what else did I say? Let's talk about length of the medians. First of all, what did I say?
This is the median. This is the median. This median has length from A to D.
How much is that length? This is also the median. How much is this length from B to E?
How much is this length from C to F? How much is this length? Every one of them has a formula for length.
median through A, formula for A to A if you want to write this one by square root of a square by b square by 4 plus c square by 4 minus a square by b square by 2 c square by 2 a square by 4 this is the length of the media or you can remember it like this half into square root of 2b square plus 2c square minus a square minus what should we put? because it goes from a, we should put minus at a Since the length of the triangle is B, we have to keep the length of the triangle minus B. Since the length of the triangle is C, we have to keep the length of the triangle minus C.
What are the lengths of ABC? This is BC, right? BC is opposite to A, because it is small a. B is an angle, right?
It has a length opposite to it, so it is called small b. This is C, right? It has a side length opposite to it, so it is called small c.
These are lengths. Triangles are side lengths. So, we can use these to find the median lengths.
The median that passes from A, B and C. The formulas related to these. If you learn one, you will understand the other two.
Okay? Right. Let us move further. Sir, what else is there related to centroid?
This is a triangle. A, B, C triangle. Now, we will take the midpoint of these two. This is the midpoint.
This is A, D. Now, what is this? If you find out AB square plus AC square, this is AB and this is AC. So, if you do AB square plus AC square, then you get 2 times of AD square either plus BD square or DC square.
Write whatever you want. Because these two lengths are same. This is midpoint, right? Since this is midpoint, you have BD and DC.
So, AD is different. So, AD square plus BD square is 2 times of. So, AB square plus AC square is nothing but 2 times of ad square plus bd square. This is the property if this ad is the median.
If this is the median, this is also a property. Any triangle, it can be any triangle. This is ad.
This is bd. That's it. Clearly, this will be used in many ways. Remember that. Now, I want to talk about the areas.
This is a triangle called ABC. This is a medium in the ABC triangle. This is a centroid G.
This is a medium. This is a medium. If I take a point G and make three triangles, I can make a triangle called AGB, AGC, and BGC.
3 will also be equal. So, in total area of the triangle, this is N1, this is N1, this is N1, this is N1, this is N1. What should I do if I want total area of the triangle? If I add this area 3 times, I get area of the triangle.
So, area of the total triangle is nothing but 3 times of the area of AGB, or 3 times of area of AGC, or 3 times of area of BGC. That's what here we wrote. Sir, you have told me about midpoints. I will tell you about that too. Let's say we are given triangle ABC.
ABC is the triangle. Now I am drawing the midpoints of this triangle. These are midpoint D, midpoint E, and midpoint F. Now if I join these three midpoints, I get four different triangles in four colors. These four triangles have the same areas.
This triangle area is the triangle. this, this and this. The areas of these 4 are also the same.
That means, in this total triangle, how much is the area of this small triangle? 1 fourth. 4 times 1 is 1 fourth. That means, what can I write as the area of the triangle ABC? I can write 4 times of triangle AFE.
Or I can write 4 times of triangle BDF. I can write 4 times of triangle DCE. Or I can write 4 times of triangle DEF. Simple.
It doesn't look like a Mughal temple. It's good, right? You can add some color to this black color.
Okay? Right. After this, what is an Incenter?
What is an Incenter? There is a triangle ABC. There is a triangle ABC.
So, I am telling you carefully about Incenter. There is an angle at every vertex. This is an angle.
This is a whole angle. Now, what does this angle do? This line goes like this. It is bisecting the angle.
What do we call this? Angle bisector. Angle bisector. Angle bisector is the angle at which the angle is half Bisecting means halfing Total angle is A and this is A by 2 This is A by 2 angle and this is A by 2 angle Similarly, angle at B is also B But half of this is B by 2 and half of this is B by 2 If angle is bisected, another line will be formed This is also angle bisector Similarly, the whole angle is half-half, this is C by 2, this is C by 2, and a line goes.
This is also called angle by sector. We are lucky that these three points are intersecting at the same point. We call that in-center I. We call that as in-center I. So this is point D.
The intersection of the internal angle by sector is point D. This is point E. This is point F.
So, from the In-Center, you can draw a line perpendicular to any side. If you draw a line perpendicular to any side, we call the length as In-Radius. And if you draw a circle using the In-Radius and the center, we call that circle as In-Circle. What do we call this circle? In-Circle.
What do we call this radius? In-Radius. So the important thing here is, the internal angle base sector from A to D is intersected with D. How much is the length of the points B and C? How much is the side length opposite to C?
It is small c. How much is the side length opposite to B? It is small b.
Since it is small c and small b, the point D in between is divided by the ratio C is to B. Take the A ratio. So, you can write the ratio as C is to B.
So, if you want the ratio near D, write C as the length on the left side of the line and B as the length on the right side of the line. After that, this internal angle base sector is a line, this is an internal angle base sector, AD. In the middle, there is an incenter I.
So, the incenter I divides the internal angle base sector in B plus C is to A ratio. Since this is a, BC should be here and A should be here. Sir, this is also there. This also intersects with the internal angle by sector. This is B and this is E.
If we divide this by A ratio, since this is B, B should be here. Since this is B, the remaining A and C should be A plus C. So, I will divide BE by A plus C is to B ratio.
So, what is the ratio here? This is C, right? So, this is C, this is E, and this is A plus B. This is divided by the ratio of A plus B E to C.
In center, these lines are the internal angle base. You have to keep it like this too. The first time viewers can also hear what one shot says. No problem. So, this is the formula for the in center.
If A is the opposite of small length A, B is the opposite of length B, C is the opposite of length C, and this is the same formula for the in center. and points x1,y1,x2,y2,x3,y3 then there is a formula for in center that is ax1 plus bx2 plus ax3 by a plus b plus c ay1 plus by2 plus cy1 by a plus b plus c is the in center formula sir, there is another x center formula right? yes there is, there are x centers in any class see what is x centers If there is a triangle A, B, C, this is A, point B, this is C.
This is a triangle A, B, C. Now, I will get 3 x-centers for this triangle. For example, if you want to draw an internal angle base sector from A, internal angle base sector is A to A. Similarly, external angle base sector is C. This is external angle base sector.
Internal angle is C and external angle is 180 minus C. If we divide 180 minus C into half, it becomes 90 minus C by 2. Sorry, 90 minus C by 2. This is also 90 minus C by 2. Similarly, if we draw an external angle at B, what will happen? 90 minus B by 2. This is also 90 minus B by 2. So, we draw an internal angle at A, and an external angle at C and B. These will also go from the same point.
We call that as first x-center. We call that X-Center and if we draw a circle by touching these 3 sides using this center we call it X circle this is X center after that we have this length we call this length as radius we call this R1 what is this R1? X radii The internal angle by sector is the same as the x-center of the incoming x-center. If I form an internal angle by sector from B1, then a circle will form here. This is called I2.
If I draw an internal angle by sector from C, then a circle will form here. This is called I3. So, for i1, i2, i3, these are the formulas for three X-Centers. Since the first is an X-Center, we have to keep it at minus of A. If it is the second X-Center, we have to keep it at minus of B.
If it is the third X-Center, we have to keep it at minus of C. Then we will get X-Centers. Right?
Right. Let us move further. What is an Arthro-Center? The formula for the angle of the centre is there but it is a waste.
Because we have to find the angle and find the centre. It is called tan2a tan2b tan2c. If we find the angle of ABC, why do we do this?
So we will do it in another method. If we find the angle of ABC, then we have to find the altitude going from A to B and C. Altitude going from A to B and height going from C. We call the intersection point of these three heights as Arthrocentre. That's it.
You should know how to find it. Because I have to find the straight line. Straight line is not known to us.
We are in 2D geometry. After going to the straight line, I will tell you. This is the 7-mark question in IP.
There is a way to find it directly. But we should know what is a line. How to find a line equation, you should know.
We can do it after knowing. So that is about the orthocenter. The concurrent point of altitude is nothing but the orthocenter. What is circumcenter?
Sir, there is a triangle called ABC. Now, I will find a midpoint for each side. DNA is the midpoint. I will place a perpendicular bisector near the DNA midpoint. It should be perpendicular and should be a midpoint.
If I find a midpoint for AB, I will draw a line perpendicular from here. Is it clear? If I find a point in AC, I draw a line from there and these three are also concurrent.
We call this as circumcenter. Then how to find this sir? I also need straight lines to find this.
But since straight lines are not told yet, I will tell you after circumcenter. But if you want an orthocenter or circumcenter at this stage, If you want it at this stage, there is a method. What is it sir? If he gives us a right angled triangle, If you give right angle triangle like this, right angle will be like this.
This is right angle, think of it as ABC. So sir, if he gives me ABC triangle, it will be 90 degrees to B. Then point B will be half center directly. Do you understand?
In a right angle triangle, where is right angle, that point is half center. and two points form hypotenuse and the midpoint of hypotenuse is the circumcenter these are shortcuts that's it so we don't need to know about other triangles I am telling about right angle triangle Babu gave three points to right angle triangle you know that is right angle triangle right angle is the center of the point Now, you know hypotenuse right angle, right? Since you know hypotenuse, call hypotenuse as a midpoint and circumcenter.
That's it. Sir, what else is the property of right angle triangle? There is Pythagoras. Pythagoras is a side square.
This is opposite to a, so it is a. Opposite to b is b. Opposite to c is small. What happened here? a square plus c square equals b square.
So, a square plus c square is equal to b square. This is called Pythagoras. If we have an equilateral triangle, Equilateral triangle means that all three sides should be the same.
If the three sides are the same, we call it an equilateral triangle. So, what is the property of an equilateral triangle? If I find a centroid in an equilateral triangle, Centroid and G Centroid is the same point, Arthrocentre is the same point, circumcentre is the same point, and Incentre is the same point.
If you find Centroid in an Equilateral triangle, Centroid will be Arthrocentre, Centroid will be Incentre, and Centroid will be circumcentre. So, the same point will be equal to 4 points. So, to find Arthrocentre in an Equilateral triangle, you don't have to do the whole process. Simply, if you find Centroid, Centroid will be Arthrocentre.
A guy gave you a triangle and asked you to find the circumcenter. What should you do? Will the three points that the guy gave you form a right angle triangle? Or will they form an equilateral triangle? You should check that.
If you form a right angle triangle, you will find the hypotenuse midpoint as the circumcenter. It will become the circumcenter. No need of any process. Shortcut. If it is an equilateral triangle, you will find the centroid.
Why? Because centroid is a formula. Easy, right?
Easy. x1 plus x2 plus x3 by 3. y1 plus y2 plus y3 by 3. So, if I ask for centroid, if I ask for arthrocentroid, I think it's centroid. If I ask for incentroid, I think it's centroid.
If I ask for circumcentroid, I think it's centroid. When there is equilateral. Okay? These two are shortcuts.
Remember these. It's very important. After that, types of quadrilaterals. There are a number of quadrilaterals. What are they?
There is usually a quadrilateral. What else? There may be a parallelogram. First, quadrilateral.
Any quadrilateral can be any way you like. It can be like this. It doesn't have a name.
Just it is a quadrilateral. It can be a parallelogram. Parallelogram means opposite sides are parallel.
Or it can be rhombus. Rhombus means all the side lengths must be equal. Side lengths must be compulsorily equal. Then rectangle. What is rectangle?
It means opposite sides are parallel. adjacent should be perpendicular to each other square should be perpendicular to adjacent and all side lengths should be equal this is it there are many types of this sir this is parallelogram right parallelogram should have opposite lengths same this length and this length should be same this length and this length should be same that is what we call parallelogram next sir, what is rhombus sir adjacent sides should be same The sides should be the same, but there should be no 90 degrees in between. What about the sides of rhombus? All the sides are the same. The difference between rhombus and square is that if you draw diagonals in the square, the diagonal lengths will be equal.
If it is A, B, C, D, AC length is equal to BD length. But it is not the same in rhombus. If it is A, B, C, D in rhombus, AC length and BD length are not equal.
AC is not equal to BD. So, diagonals are not equal in Rambas. But, how are the diagonals in square? They are equal. Sir, you said that opposite sides are parallel.
What is the difference between rectangle and parallelogram? In parallelogram, diagonals are not equal. But, they are equal in this.
They are equal in rectangle. Do you understand? So, if all sides of diagonals are equal, Square. Only, if all the sides are same and the diagonals are not equal, it is rhombus. If the opposite sides are parallel and the diagonal lengths are same, it is rectangle.
If the opposite sides are parallel and the diagonals are not equal, it is parallelogram. If nothing is there, it is just a quadrilateral. You have to remember these.
Okay? Right. Very good evening, Amma.
Missing vertices. Babu, you are in this. For example, we are asked a lot of questions in parallelogram. What we are asked in parallelogram is, a of, sorry, x1, y1, b of, c of, then what is d?
What is the d point? This is parallelogram. This is parallelogram and the d point which is missing is asked. Then what should be done is, diagonals bisect. What happens when diagonals bisect in parallelogram?
AC has a midpoint and BD has a midpoint. That means, midpoint of AC is equal to midpoint of BD. So, AC has a midpoint and BD has a midpoint.
Both are same. So, what is the midpoint of this? x1 plus x3 by 2 comma y1 plus y3 by 2. This is the midpoint of this. What is the midpoint of BD? x2 plus x4 by 2 This is x4 comma y4 comma y2 plus y4 by 2 Equal these two Both are the same Equalize the values of x4 and y4.
What is the value of x4? x1 plus x3 minus x2. What is y4? y1 plus y3 minus y3. That is nothing but x4 comma y4.
This is the fourth point in parallelogram. If you ask about missing vertex, diagonals bisect the same point. If you remember like this, there will be no more problems. Logically, we should remember that the reason for this formula is because of the diagonals are bisecting at the same point. That means the midpoints of the diagonals are same because of that.
After that, what is locus? It is very important. From JEE, he removed the transformation and translation of axes.
No, he removed it from 2D geometry. So, I will only explain locus. There is nothing to explain locus.
I will explain the small point locus. But we have to solve a number of questions on this. I will buy it, you also buy it. Sir, we are not going to directly question after this. Now we are going to do some questions.
Sir, what is a locus? Locus means that many children have a lot of doubts. Now, what Kiran sir will do is, he will grind your doubts into a mix and then throw it away.
If you have so many doubts, let's grind them into a mix and throw them away. Okay? Right. For example, there is a question.
The question is given by the instructor. You are the point. Student is the point and Kiran sir is the question. Kiran sir is telling the student.
First, the student is here. He is telling the student to go straight for 5 km. Sir is telling him to go straight for 5 km.
After 5 km, I told him to turn left and travel 2 km. He turned left and traveled 2 km and stopped. I told him to travel 4 km to the right.
He turned right and traveled 4 km. He stopped. The student who was supposed to be here, came here. How did he come here? How did the student come here?
By obeying my conditions. By accepting the information I am giving him, the student travelled in accordance with it. First he went this way, then this way, then this way and then this way.
So now when he is going, he has created a path. This is the path. He went this way, then he went this way.
and then he went to Eidara. So he was obeying the conditions I told him and created a path to travel. This path is called locus.
That locus is the locus of the point. You are the student of the point. Then, The person who listens to and obeys the words of others is Kiran sir. What is Kiran sir? I am a question.
He gives information in questions. He says go left, go right, go straight. That is the information.
Listening to all that information, You are the point. So, point is traveling by listening to information. When it is traveling, it creates a path. If we write that path as an equation, we call it locus. We call this path locus.
Then, where is this path? We have to tell an equation to this path, right? We have to tell an equation, right?
We will tell a mathematical equation. We will call that mathematical equation locus. But, actually, do you understand locus or not?
Do you have any doubt? That means... We call the path that obeys the conditions given in the question and goes, and the path that obeys the point and goes, that is called the locus of the point.
We should do this mathematically. What points are there for this? What do you have to keep in mind and go, such that I will get the correct answer.
First of all, why? What do you do? What point?
It means, find the locus of a point. Find the locus of a point. Let that point as h,k. Let point as h,k.
Or alpha,beta. Or a,b. Think of any one. Or m,y.
It is your choice. h,k,alpha,beta,a,b. Think of any one. But that point should be considered as any one, not x,y.
Second, use. The information given in the question Use the information given in the question Eliminate x and y in the equation If you want to know the answer, you should not have x or y in the answer. You should have h,k in the answer.
h square, h cube, h by k, k square, y, h, h, k, x, k, y, x square, h square, x, y should not be there in the answer. If it is there, you will not get the answer. What to do? I have to eliminate x and y.
Try to eliminate the questions with h,k. So, in the final answer, replace h,k by x,y. Finally, you got the answer in h,k.
In that answer, remove h,k and replace x,y. If you want to follow this, you need to follow these 4 points. Okay sir, let's do some questions. For example, there are many types of questions. Some people ask directly.
For example, find locus of P such that such that PA is equal to PA by Pa is to Pb is equal to 2. What does Pa is to Pb mean? 2 means 2 is to 3. 2 is to 3 means, where P is a locus, where A is nothing but 1,2, B is nothing but minus 1,3. Like this. Now, what I will do is, I have to use the information that he gave me.
What did he give me? Information is, PA by PB is equal to 2 by 3. This is the given information. He said to use the entire information.
I am using what he gave. What should we think of P? Let P as H, K. Think of P as H, K. PA is the distance between P and A.
The distance between A and P. So, this is PA. 3 times of PA is equal to 2 times of PB. What will you do now? I will do the squaring on both the sides. Why, sir?
Now, if I find out PA, what will be the result? It will be square root of... Distance formula is square root, right? What will I do to avoid that square root?
I will do the squaring on both sides. 9 PA square is equal to 4 into Pb square. Now, PA is square root, right?
It will be divided by square root and 9 times... So, PA is the distance between point P and this. What is that? h minus 1 whole square plus k minus 2 whole square.
is equal to 4 times of Pb, distance between P and B. This is distance between P and A, Pb. So what happens here is, h plus 1 whole square plus k minus 3 whole square. Solve this.
If I solve this, will the complete answer come in h and k? The complete answer will come in h and k. If I do it in IP, I can do it. Now what we will solve is, 9xh2-2h which is 18h plus 1 which is 9 plus k2, k2 is 9k2 minus 4x36k plus 4 is 36 is equal to, we have to do here 4h2 plus 4 plus 8h plus k2 minus 6, 24k plus 9, 36. So, here 36, here 36, cancelled. So, 9h square, 4h square, what is left?
5h square. Here, 9k square, k square means 8k square. After that, minus 18h plus 8. Minus means 26h.
Here, minus 36k plus, means 12k will come. Minus 12k. What are the constants here? 9-4 is 5. 5 is equal to 0. So, we have H2O2.
Now, we have to write Hdx and Kdy. 5x2 plus 8y2 minus 26x minus 12y plus 5 is equal to 0. That's it. That's it, friends. This is what we call locus. This is what we call locus.
There is nothing. There is only this much. Solving the process. Sir, do you always do it like this?
Because locus is You can ask in straight lines, circles, any geometry question, locus, even complex numbers. But, since we have learnt 2D geometry, we will be asking questions in 2D geometry. If you want to ask in higher level, let's do PYQ. We will be solving the questions that were asked in 2D geometry in previous year.
Ready? Right, we still have half an hour left. Let's do it fast. The question is, A triangle has the vertices 1, 2, 1, the midpoint of two sides through it are midpoints of the sides. Then, the centroid of this triangle, very simple, right?
Can you understand what I asked? This is a triangle ABC. A is 1, 2. We don't know what B is and C is not known to us.
But he gave a midpoint to AB. That is. AC is given a midpoint of 2,3. Now, we have to find out what the centroid is. What is the formula to find the centroid?
x1 plus x2 plus x3 by 3. y1 plus y2 plus y3 by 3. But there are no two points here. How will you find it? 5x square. We have already found 5x square. Is it clear?
5x square. Sir, this is the point. If b is x2 and y2, I don't know this.
This is x3, y3. So, this is the midpoint of a to b. What is the midpoint formula for these two?
What is the midpoint for this? x2 plus 1 by 2. x2 plus 1 by 2. What does that mean? It must be minus 1. y2 plus 2. y2 plus 2 by 2. What does that mean?
It must be 1. So, from here, how much will x2 be? It will be minus 2 minus 3. How much will y2 be? It will be 0. What does this point mean?
minus 3 comma 0 similarly, this is the midpoint for these two too so, what we are doing is x3 plus 1 by 2 is equal to x3 plus 1 is 4 x3 is 3 comma y3 plus 2 by 2 is 6 this is 6 minus 2 is 4 so, 3 comma 4 is the C point since I got ABCs, let's use the centroid formula G formula x1 plus x2 plus x3 by 3... Y1 plus Y2 plus Y3 by 3. 6 by 3 means 2. So, 1 by 3 comma 2 is the correct answer. B option.
Any doubt? Clear? Very good.
Very good, Sreeja. Super. Let us move further.
Let's move to the next question. Let's move to the next question. When did he ask this?
He asked this question in 2011. What are the simple questions? He asks simple questions from 2D geometry in 2011. If in a parallelogram ABCD, the coordinates of ABC are represented... Then equation of the diagonal AD.
What is the diagonal AD? For example, this is parallelogram. This is parallelogram ABCD. In this, ABCD and R respectively. ABDC.
CD is the move. He gave DC as he wanted. He said ABDC.
I am taking ABDC. So parallelogram, the coordinates of AB and CR respectively. A gives B, C gives A.
A is 1, 2. B is 3, 4. D is 2, 5. C is 2, 5. D is like this. Now what he is asking is, do I want diagonal AD? He wants AD diagonal.
He wants equation. So this is actually a straight line equation. But we can find out.
What this means is, if I find the midpoint of B and C, the BC midpoint is on the AD line, right? Yes, sir, it will be there, right? So, BC midpoint is enough. What is BC midpoint, sir? 5 by 2 comma 9 by 2. That means, our line should go from these two points.
From A point and from the diagonal midpoint which is here. The line equation from these two points is enough. What is the equation of line?
You may not know. I have to tell you actually. I will tell you.
It is equal to slope into x minus 1. What is the slope, sir? Y2 minus Y1 by x2 minus x1. 9 by 2 minus 1 by...
5 by 2 minus 9 by 2 minus 2 by 5 by 2 minus 1 so y minus 2 is equal to 4 and a 5 by 3 3 times y-2 is x-1. So, this is 3y-6 is equal to 5x-5. So, this is 5x-3y plus 1 is equal to 0 is the answer. 5x-3y plus 1 is equal to 0. C option is the correct answer.
I will tell you straight lines tomorrow. But, is it very easy or not? There is no coordinate geometry here. He is asking about diagonals and parallelograms. What is the only thing we have used here?
Diagonals bisect. Diagonals are bisected. Okay. Question number 3. If a straight line passing through the point minus 3 comma 4 is such that its intercepted portion between the coordinate axis is bisected at P then its equation is what?
Look carefully. This is what Locus is asking. If the straight line passes through the point then the coordinates of P are bisected.
This is If the straight line passing through the point is such that It's intercepted portion between the coordinate axis bisector equation. No, no, no, no. This is related to locus, but this is not related to locus. This is related to straight line. This is related to straight line.
We are not doing this now. If we do this straight line, we will bring it tomorrow. If we do straight line tomorrow, we will bring it tomorrow. There is no locus anywhere.
I thought there was locus, so I gave it. This is not a locus question. Okay, look at the fourth question. A straight line through a fixed point intersects the coordinate axis at the point P and Q. If O is the origin and the respected OPQ is completed, then the locus of R is what?
R is a locus. Straight line through O is a fixed point. A line goes from, intersects the coordinate axis at distinct points P and Q. For example, this is 2 units, 3 units. This is.
A line goes from here to here. This white line intersects us at two points. These are P1 and Q1. It is at PQ.
And OB is the rectangle. OP, R, Q. Q is PQ. completed then the locus of R. O is origin, P is OP, RQ is completed then the straight line rectangle. OPRQ means the rectangle should be here.
He says, this r is a locus. We need a locus for this. What do we need for this locus?
We should think of it as h, k, or k. I thought of it as h, k, or k. Now, what I will do is, I will use all the information given here and change this locus.
When the straight line changes like this, the rectangle r point also changes like this. He is asking what the locus is. What can we do for this?
To find this simply, What is this point? This point becomes. This is, this point is. This point is.
It becomes. Or, let's assume it is. Since it is the midpoint of the two, this point is directly coming, right?
Not. Let us take this as... midpoint is 2,3 for example, this point is 0,B now this is this is this is fixed point this is not the midpoint this just it is a point this is just a point p is a point 2,3 is also a point so just it is a fixed point it is not the midpoint so I want the locus for h,k so let's do it directly now I thought this is h,k this is 0,k so what is this line equation x by h plus y by k is equal to 1 but this point goes from line 2,3 so we can substitute 2,3 so it is 2 by h plus 3 by k is equal to 1,5 that is the locus we need we got the answer in h k now I will take LCM LCM is 2k plus 3h By hk is equal to 1. From here, this is 3h plus 2k is equal to hk.
So, after hk comes, we have to remove hk and put xy. So, 3x plus 2y is equal to xy. Is it there or not?
See. 2x, 3x plus 2y, yes. C option is the correct answer.
I think this is midpoint. It is not midpoint. It is a fixed point. Fixed point. I am trying to take it in the middle and try it like that.
But it is a fixed point, not a midpoint. If it is a midpoint, it will get the same answer. It will get the same answer. It will not get a locus.
So, this is an eneme point. It means that the point can be here, here, or here. Okay? Right.
So, this is the way to find out the locus. What I am saying is, the point at which I want to find the locus, that point is. If this is, then its foot is.
And its foot is. These two are. Line equation means, the line equation from this point to this point, x by h plus y by k is equal to 1 by the intercept form.
So, this line goes from this point, so if we substitute the points, we get an equation in hk, hk is xy beta. That is the locus of the point we need. Okay? Right.
These are all PYQs, ma. Questions asked in previous years. When did he ask?
He asked in 2018, I guess. Come on, after that. Let K be an integer such that the interval with the vertices.
Has area 25 square units then the Arthrocentre of this triangle is at the point Where is it? Then the Arthrocentre of this triangle is what? At the point If you want Arthrocentre formula, you have to Stratelize it So, I will rape this question because what I said when I told you that there is no formula for the orthocenter I said that we have to find the altitudes we have to find the altitudes and the intersection point of the altitudes is the orthocenter straight line passing through the point having the slope we have to find that so we have to solve this like that but if this is a right angle triangle if this forms a right angle triangle the right angled point will be the orthocenter But I know that it is not the case here. But let's try it.
I can answer only if it is right. Or else you will have to do straight lines. First of all, I am going to find out the k value.
What do I think the k value is? Half into, I think this is the area. What is the area?
k-3k1, 5k1-k21 What is the area? It is 28. Here we have a model s, if we remove the model s, right side will be plus or minus 28. So this is 28 into 2, 56 is 56 which is plus or minus. Is equal to, let's calculate, here what is 1 into, del to the row, del to the column, ad minus bc or k into, we can do it like this, k into del to the row, del to the column, ad minus bc, k minus 2. So k into k minus 2, minus or minus, plus 3k into del to the row, del to the column, ad minus bc, 5 plus k.
plus 1 into, delete the row, delete the column, AD minus BC, 10 minus of minus plus k square. This is how it came. But since it is modulus, we will remove the plus or minus. This is minus r k square minus 2k plus 3k into 5, 15k plus 3k square plus 10 plus k square, sir.
What is this, sir? This is. So, k square, k square, 5k square, sir. 5k square, sir.
Here, 15k is equal to 13k plus 10 is equal to plus or minus 56. Now if we take this left side, it becomes a quadratic. When you want to find the values, you need to check the K value. What I said earlier is that we need to find the points only.
You need to find this line and this line. You need to find the lines and find the intersection point. It will come in a big number, so I am not doing this here. So, you know how to find the K value, right?
You need to find the K value and use the K value to find the points. You need to use these points to find the Arthrocentre. After you get the ABC points.
That is related to straight lines. That is why I told you later. So, you are asking less from 2D geometry and asking less from other things.
Next question number 6. Question number 6. Let P is the point, Q is the point, R is the point, B is the point, then the equation of the bisector of the angle PQR. He is asking what is the angle bisector of PQR. This is P point, 0 point, this is 0, 0 point. This too should go straight lines.
Angle bisector. Again straight lines are required. He is not directly asking us the question.
This is, P point is, and R point is here. This is. So, this is R point. Where do we need these R points? P, Q, R. This is Q.
So, P, Q, R means, here you also need angle by sector. If we draw a triangle, we can write the angle by sector here. You can write the line equation for this point. I told you that this point divides the ratio of R and P.
I told you that this length divides the ratio of E length. E length is 1. E length is square root of 9 plus 9 th result is 27. 36 is 6. So, this point is 6. This point is 1 is to 6. This ratio. This is the length ratio and that is the length ratio.
This point D is divided by 1 is to 6 ratio. So, what is the point divided by 1 is to 6 ratio? We can find it in the section formula. We have to do cross multiplication. Let's do it now.
Cross multiplication means, this is this, from this, 3 is given, from this, minus 6 is given by 7. Comma. From here, 3 root 3 comes, from here 0 comes by 7. Because 1 plus 6 is 7. So, the point is minus 3 by 7 comma 3 root 3 by 7. So, from this point, 0 comma 0 passes. What is it? y is equal to m x.
So, y is equal to m. This is m. What is m? y by x.
That's it. y 2 minus 0 by x. So, what we get is 3 root 3 by 7 by minus 3 by 7 into x. 7, 7 gone.
3, 3 gone. Left with minus root 3 into x. That's it.
This is the correct answer. If we bring this to the left side, root 3 x plus y is equal to 0. Root 3 x plus y is equal to 0. This is the C option. C option is the correct answer. This is also PYQA.
ER is not here. But this is also... previous year question. Here, I used 2D only to get the D point. But, this line is Y is equal to MX line.
If you think of the slope in this line, you will get the answer. You will have to use the line. If you want to give a complete coordinate geometry, you will have to use it. Some are less. I have told you the first, second, third questions.
You have only 2D. We will not go further. But here, he is also trying to find the straight lines. Here, we have to find the straight lines.
Right. After that, there is no question number. Right.
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Which grade are you in? 12th class. Which grade are you in? JEE Choose of Mode Exam is offline Which city are you in?
Hyderabad or Vijayawada What is your preferred date? 3rd December I will tell you a good news now I am going to tell you a good news For mobile number 1,2,3,4,5,6,7,8,9,10 After hitting the number There is a register for free You will get a verification code Enter the code and submit the code That's it registration is done Sir, you are saying good news Sir, you have freebies I will tell you what they are Listen carefully I told you that there is a chance to come in the evening. I think it has come. Now we can see that there are freebies.
It has come to us. I'll show you what it is now. What is freebies?
It's not like that. Everyone who has written the exam has a gift. You asked me earlier. I will tell you why we discussed this and why we got these evolves.
Openly. So what do you say in general? Oh God! It is open.
So generally, all the kids are ready. Sir, we have applied online and we have done offline. We are registering online and writing about Vijayawada, Hyderabad or Vizag. We should have a difference between those who write online and those who write offline.
Online is like writing a book and sitting at home. Offline is like traveling a little further. We have 30-50 km and we are writing that far. What we want is that those who are writing offline, we should give them extra information. Right, they haven't given this yet.
This is not updated yet. Assured gifts for all offline test trackers. Yes, what did he say? Every single kid who comes to write an offline exam, girl or boy, everyone, assured gifts for all offline test takers. Those who are going to come to offline center and write an offline exam, even those kids, There will be compulsory and assured gifts.
This is what we have just received and it has been updated. So, I will tell you about the gift tomorrow. The update is not yet available.
I will tell you about the actual gift tomorrow. Sir, I told you earlier that you should not write offline in one center. Write one on the 3rd and one on the 10th. Give two numbers.
If you are in Hyderabad, you are in Vijayawada, you are in Vizag, write two numbers. You can go twice and get a gift twice. You can also get a scholarship.
You can also do 5 days space research. There is also a 5 days trip. It is fully paid. They will pay.
You don't have to do anything. So, if you have all these offline, you can also get extra gifts. That is for encouragement.
We are coming so far for encouragement. Our kids asked, so we asked. They said, sir, we will update tomorrow. Earlier, when I came, I couldn't update. Now, it has been updated.
So, there are also assured gifts. Definitely try it. And write two or three exams with your father's mobile number and one with your mother's mobile number and one with your mobile number. Write two online and two offline exams and get a gift in between.
Okay? So first of all, try to get a 100% scholarship. Okay?
So that's all for now guys. Actually, I have a paid class now. I have an M.S.Ed class. So I have to go early for that. I already told you, every Friday I will be taking earlier classes.
I told you that I will be taking 5K instead of 6K. So, I will be taking live classes on Friday. Okay, guys. That's all for now, guys.
Bye, everyone. Thank you all. Bye, all. Meet you all. Thank you.