How to Identify a Conic Section by Its Formulas

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  • 0:02 A Conic Section
  • 0:56 Circles
  • 1:39 Ellipses
  • 2:35 Parabolas
  • 3:33 Hyperbolas
  • 4:45 Lesson Review
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In algebra, we have a formula for everything. Take a cone and slice it and we have a formula for the shape that results. Watch this video lesson to learn how to distinguish between these formulas.

A Conic Section

Picture an ice cream waffle cone right side up. Now picture another one directly underneath it that is upside down. Now take a knife and make a cut through it. This is what we call a conic section, the shape that results from cutting though a cone.

Try cutting it in different ways. Notice how the shape that results changes depending on how you cut it. If you cut it straight across, you get a circle. If you cut it at an angle, you get an ellipse, which is a circle that has been stretched out. If you cut it parallel with the edge of the cone, you get a parabola, which looks like an arc. You get a hyperbola, or two arcs, if you cut so that you go through both the top and bottom cones.


Each of these shapes has a unique formula that identifies it. You will learn how to identify these different equations in this video. Let's get started now with circles.


We get circles when we cut our cone straight across. The formula for our circles has a standard form of


An example of a formula that will give us a circle is x^2 + y^2 = 1. Notice how we have the x^2 part and the y^2 part as well as a plus separating them. We also have the radius part, which is always positive. If you see these, then you are looking at the formula for a circle.


The formula for an ellipse, which is a stretched out circle, is almost the same as a circle. The standard form for an ellipse is


An example of an ellipse is (x - 1)^2/4 + (y - 4)^2/9 = 1. What we look for here is the x^2 part with a number underneath and the y^2 part with a number underneath. If one of the radiuses is 1, then you won't see a number there for that radius. We are also looking for the equals 1 part as well.


Next comes our parabola, which is an arc. The standard form for our parabola is


An example of a parabola is x^2 = 8(y - 2). Our identifying point for a parabola is that if our x is squared, then our y is not, and vice versa.

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