After the straight lines, the conics are the next most important class of loci. There are many usage and applications of conics in calculus, and indeed throughout all mathematics. Conics are the part of some high school curriculums, but not others, so I expect that some of you have seen this before, but for others this is new material.
In this video, I'll try to provide a good conceptual understanding for conics, as well as show you their equations, so that you can work with them algebraically. The term conic is short for conic section, which means a slice of a cone. Conics come from slicing cones, and look into the shapes you get in the cross section.
There are four possibilities. A perfectly horizontal slice of a cone will give you a circle. A slightly tilted slice will give you an ellipse. A slice precisely at the same angle of the cone will give you a parabola.
And a slice steeper than the angle of the cone We'll give you a hyperbola. Geometrically, slicing a cone is how to understand conics as a unified family of shapes. They all come from this same source. Algebraically, they are also unified. For equations of lines, I allowed the variables x and y multiplied by constants, but no other operations.
For conics, I allow x squared, x times y and y squared, but no other new operations on the variable. Again, I can multiply by constants and get expressions like ax2 plus by2 plus cxy in the equations, and then by adding and subtracting these kinds in equations and looking at the loci, I get conics. After lines, these terms have the most basic algebraic operations, so it makes algebraic sense that conics are the next set of shapes to consider. Now I'll go through the four conics individually, describing their shapes and equations. The first conic is the circle.
As a slice, it is a perfectly horizontal slice. It's this slice you can see in the diagram. Geometrically, a circle is all points which are a fixed distance from the centre. Algebraically, if I let r be the radius, and make sure the circle is centred at the origin, the equation of the circle is x squared plus y squared equals the radius. square.
The second conic is the ellipse. As a slice, it's a slightly tilted slice, as you can see by this slice in the diagram. Intrinsically, an ellipse is similar to a circle, but instead of a single radius, the distance from the centre varies.
I can describe an ellipse using two distances, the closest point to the centre and the furthest point from the centre. The distance to the furthest point is called the semi-major axis, and is represented by a in this diagram. The distance to the closest point is called the semi-minor axis, and is represented by b in the diagram.
These two numbers also show up in the equation of the ellipse as denominators. The general equation is x squared over a squared plus y squared over b squared equals 1. In this diagram the larger distance is horizontal, since a is larger than b. This doesn't need to be the case.
If b is larger than a, then the ellipse would be taller than it is wide, and the vertical distance, the larger distance, would be the semi-major axis. Semi-major doesn't necessarily mean x or y, a or b. Instead, semi-major means whichever distance is the largest, and likewise, semi-minor means whichever is the smallest.
The third conic is the parabola. Other than the circle, this is probably the most familiar, since the parabola is also the graph of a quadratic. As a slice, the parabola comes from a slice that perfectly matches the angle of the cone. It's this slice in the diagram, and by sliding over I can see that the slope of the slice is exactly the same as the slope of the edge of the cone.
If it is centered at the origin, the equation y equals ax squared. The parameter a controls how wide the parabola is. If a is large, the parabola is narrow, and if a is small, the parabola is wide. As well, if a is negative, the parabola will open downward, but if a is positive, the parabola will open upward, as in this diagram.
The most general form of the equation of a parabola is y equals ax squared plus bx plus c. Here you can see again that this is the graph of a quadratic, f equals ax squared plus bx plus c. Some of you might be familiar with another form of the parabola, the vertex form.
I can also write the equation of a parabola as y equals a x minus b squared plus c. In this form the geometry is a bit clearer. The point of the parabola called the vertex is at the coordinates. The constant a, like in y equals a x squared centered at the origin, controls the width of the parabola and whether it opens upwards or downwards. To write a parabola in vertex form Uses an algebraic process called completing the square I'm not going to review that process in this video But it is a useful process that you should be familiar with Finally the last conic is the hyperbola as a slice of a cone It is a slice that is steeper than the angle of the cone It is this slice in the diagram the equation of a hyperbola looks very much like the equation of ellipse The only difference is that the sum is now a subtraction The x squared over y squared plus y squared over b squared equals one of an ellipse becomes x squared over a squared minus y squared over b squared equals one for hyperbola.
The interpretation of the numbers a and b is not quite as an immediate for hyperbola as it was for an ellipse, where the larger was the semi-major axis and the smaller the semi-minor. However, the arms of the hyperbola, as they get further and further from the origin, become closer and closer to straight lines, shown by the dotted lines in the diagram. The slopes of these lines are a over b and negative a over b, showing a bit how the numbers in the equations affect the shape.
This concludes the very quick definition and review of conics. As I said before, conics are useful and ubiquitous. One of the most important applications of conics, particularly historically, is astronomy.
In ancient Greek mathematics it was assumed that orbits of planets and moons were circular. The Greeks developed quite a complicated system of circles on top of other circles called epicycles to match the data of objects moving through the night sky. Johannes Kepler in the 16th century proposed that orbits are not circles, but more generally any conic.
This proved to be correct. at least in approximation, in complicated systems of multiple moons and planets. The orbits are still roughly conics, but have finer details, of course. Conics are still used as the basic path for orbits. For objects with high enough velocity to escape the gravitational pull of a larger body, called the escape velocity, the orbit is hyperbola.
This shape is open, it only makes one pass before it flies away forever. For objects where the velocity is lower, they are captured into a repeating orbit, and these orbits are ellipses, which are closed and therefore repeat, going around and around the central object.