[Music] good day everyone welcome to your free health christmas i am mr bell and i will be your customer [Music] pre-calculus has three main parts the analytic geometry the series and sequence and trigonometry of course we are going to start with the analytic geometry can academy define analytic geometry as the coordinate dramatic here objects are visualized in a coordinate plane points lines curves and planes are dealt with using algebraic methods the very first topic that we are going to discuss under the first part of pre-calculus is veconic sections but before we discuss all about conic sections let us first try to find out how when or where this conic section originated the history of chronic sections may be traced back to ancient greece at the time of the greek mathematician minecraft at the time king minus wanted to build a tomb but thought that the current dimensions of their tomb was too louse king menus wanted to double the size of the cube without affecting the length so several mathematicians tried to find a solution but only manet moose was able to provide one he was able to discover the conic sections from here but at that time the conic sections were named differently from what we know now those conic sections were named the section of a right angle cone the section of an obtuse angle cone and a section of an acute angled cone after mineck was introduced the idea and concepts about conic sections many other mathematicians investigated it including euclid and aristelius although the next major contribution for chronic sections was done by archimedes and apollonius [Music] the great archimedes never published a book that is solely about the chronic sections however he was able to publish books which contains some of the concepts of chronic sections and those are the quadrature of parabola colonoids and spheroids floating bodies and plain equilibrium although there are no evidences of the work of euclid it is said that he inspired the work of the great archimedes apollonius who was known as the great geometer was able to publish eight books that is solely about the conic sections the first four books were in greek translation the next three books were in arabic translation and sadly to say the last one was lost entirely apollonius was also the one who proved that all this conic sections may be derived from one and the same cone he was also the first one to use the double napped cone apollonius also gave the names of the conic sections which are used up to now the parabola the ellipse and the hyperbola although it was believed that all of the work of apollonius was derived from the work of the great archimedes after apollonius papua's and proflus shed light to the contributors of the conic sections they were the reason why previous mathematicians were credited for their works and they did this by providing commentaries on different works in the 5th century it was until the 15th century around the era of renaissance when the interest in rape culture and the greek knowledge arose and at that time at 1605 johannes kepler introduced his idea that mars revolves around the sun in an elliptical pattern this discovery fueled motivation to study chronic sexuals although it focused more on astronomy kepler was also the one who discussed that instead of three we have five conic sections and those are circle parabola ellipse hyperbola and lines he also stressed out that the parabola and the lines are the extremes of ellipse other pioneers to the discovery of conic sections were newton [Music] [Music] that concludes the origin of the conic section now let's go with its definition conic section is a set of points whose distances from a fixed point are in constant ratio to their distances from a fixed line to derive the conic sections we will be needing a double knot cone and a cutting plate the conic sections have a general equation a x squared plus b x y plus c y squared plus d x plus e y plus f is equal to zero there are two types of phonics sections the degenerate conic sections and the non-degenerate conic sections let's start with the generated phonics this conics are formed when the cutting plane passes through the vertex of the double napped cone we have the point line and two intersecting lines now let us take a look on how this are formed by using a double napped cone and a cutting plate first let's have the point the line [Music] and the intersecting lines we are actually going to focus more on the second type of the conic section which is the non-degenerate chronic sections these conics are formed when the cutting plane does not pass through the burnings of the double knot tone we have the circle the parabola [Music] the ellipse and hyperbole the circle is formed when the cutting plate is parallel to the base of the cone let's take a look at how it is worn some examples of circle are a wall [Music] a circular plate a circular table [Music] and many more next we have the parabola the parabola is formed when the cutting plane is parallel to an edge of the cone let's take a look on how it is formed some examples of parabola is at the arc of the mcdonald's logo a banana a satellite dish and many more an ellipse on the other hand is formed when the cutting plane is not parallel to any of the edge of the cone let's take a look on how it is formed [Music] some examples of the ellipse are the egg the football the orbit of the earth and many more the last conic section that we have is the hyperbola the hyperbola is formed when the cutting plane intersects the two knobs of our double knot cone let's take a look on how it is formed [Music] some examples of the hyperbola are the hourglass of course the bastical chord and many more as defined earlier conic sections have a constant ratio and this ratio is known as the [Music] eccentricity eccentricity as defined by milan and al is the ratio between the distance of a fixed point called the focus and the distance of a fixed line called the directrix from a set of points [Music] the parabola has an eccentricity of one an ellipse has an eccentricity that is less than one the hyperbola has an eccentricity that is greater than one while the circle which is considered as a special kind of ellipse has an eccentricity of zero once again this is miss bell echoing to you what euclid once said the laws of the nature are but the mathematical thoughts of god