# 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.

## Circles

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

hkr

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.

## Ellipses

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

hkrxxryy

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.

## Parabolas

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

hkphkphpk

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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