Did you know that if you slice a cone in different ways, you can make a circle, an ellipse, a parabola or a hyperbola? And, did you know that you can express all these shapes with just one general equation?
Imagine one of those bright orange traffic cones that you see on the road. Now, if you were to slice through that single cone at different angles, you'll see that you can end up with several different unique flat shapes. You get circles, ellipses, and parabolas. And if you glue two traffic cones tip to tip and slice through both, you'll get a hyperbola. The shapes that you get from slicing a cone are called conic sections.
In the picture above, the number one cone is the parabola. It looks like a u-curve. In number two, the bottom slice gives you a circle, a round shape where any point on the edge is the same distance to a central point. The top slice in number two is the ellipse, which is an elongated circle, or an oval. The slice in number three cuts through both cones straight down, and this gives you a hyperbola, two equal parabolas that look like mirror images of each other.
And of course, because this is math, you have formulas for all of these. This is the general formula for conic sections that covers all of your slice shapes:
In the formula, A, B, C, D, E, and F are all constants. This general form covers all four unique flat shapes. The values of these constants help to determine the shape. But you can't use the general form to determine the shape. You'll only know that by changing the general form to a standard form by completing the square for both the x and y variables.
To complete the square, rewrite the general form so all the x terms are together and all the y terms are together. Move the constant to the right side of the equation. Then complete the square for the x terms and the y terms, but remember to keep the equation balanced. Whatever you add or subtract on one side, you also have to do to the other. So, if you add 100 on the left side, you also have to add 100 on the right side. Then do some more math (divide everything by the constant on the right) so you end up with a one on the right side. You may want to change the denominators on the left side to be expressed as squares. Once you have the equation in standard form, you can determine whether it's a circle, ellipse, parabola, or hyperbola.
Here is an example of completing the square to convert from the general form to the standard form.
Now that you've changed from general form to standard form, you can figure out what kind of shape it is.
If it's a circle, then the standard form will follow this form:
The center of the circle is at the point (h, k) on the coordinate plane. The radius of the circle is r.
Because the circle has the radius as the denominator for both the x- and y-values, you can write the standard form for a circle in two ways, by either keeping the r term in the denominator on the left side or moving it to the right side.
Now, if you compare the equation you just changed (from the general form to the standard form) to the standard form for a circle, you see that it is a circle. It follows the standard form for a circle.
The standard form for an ellipse follows this form:
The center of the ellipse is the point (h, k) just like with the circle. Since the ellipse is an elongated circle, the a tells you how long your ellipse is in the x direction if you multiply it by 2 (2a) and the b tells you how long your ellipse is in the y direction if you multiply it by 2 (2b).
A parabola that opens up or down has this standard form:
The vertex of the parabola is the point (h, k). The p will never equal 0. The focus of the parabola is located at the point (h, k + p). Its directrix is given by the line y = k - p. And, the axis of the parabola is the line x = h. Parabolas can also open left or right. That equation exchanges the x and y and the h and k.
For a hyperbola that opens up or down, the standard form is this:
The center of the hyperbola is the point (h, k). Since the hyperbola is made up of two parabolas, it will have two vertices and two foci. Similar to a parabola, a hyperbola can also open left to right. That equation also interchanges the x and y and h and k.
The key to figuring out which shape you have is to look at where the x and y terms are and what's being added or subtracted from what. Also, look at the denominators to see if the a and b are the same as for the circle. If they are, then they're the radius of your circle.
Let's take a couple moments to recap what we learned in this lesson about identifying conic sections. A conic section is the flat shape you get when you slice a cone. There are four unique flat shapes.
The general form equation for all conic sections is:
In order to figure out what shape you have, you need to complete the square and see which standard form equation matches your equation.
The standard form equation for a circle, or a round shape where any point on the edge is the same distance to a central point, is:
The standard form equation for an ellipse, an elongated circle or an oval, is:
The standard form equation for a parabola, a u-curve, is:
The standard form equation for a hyperbola, which is two parabolas back to back, is: