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Identifying Conic Sections: General Form & Standard Form

Identifying Conic Sections: General Form & Standard Form
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  • 0:04 Conic Sections
  • 2:31 Circle
  • 3:15 Ellipse
  • 3:43 Parabola
  • 4:17 Hyperbola
  • 4:59 Lesson Summary
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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.

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?

Conic Sections

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.


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:


conic sections


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.


conic sections


Now that you've changed from general form to standard form, you can figure out what kind of shape it is.

Circle

If it's a circle, then the standard form will follow this form:


conic sections


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.

Ellipse

The standard form for an ellipse follows this form:


conic sections


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