Practice with the Conic Sections

Practice with the Conic Sections
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  • 0:02 What Are Conic Sections?
  • 0:34 Circles
  • 1:15 Ellipses
  • 2:42 Parabolas
  • 3:53 Hyperbolas
  • 4:28 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Conic sections are shapes created by cutting through a 3D cone. In this lesson, learn how to identify each conic section from its graph and characteristic equation.

What Are Conic Sections?

What do shapes like circles, parabolas, ellipses, and hyperbolas all have in common? They are all created by the intersection of a flat plane with a three-dimensional cone and so are called conic sections. Conic sections have been studied since ancient times and appear in nature in many contexts, from the shadow cast by a light to the orbit of a planet. Let's look at each of the conic sections and see what they look like and how to represent them mathematically.

Circles

A circle is the shape that you would get if you cut the cone straight across at a right angle to its axis. The formal mathematical definition of a circle is the set of points that are all the same distance from another fixed point. The fixed point is the center of the circle, and the distance from the center to every other point is the radius.

If the center of the circle is at the origin of the coordinate system, then the equation that describes the circle will be:

x^2 + y^2 = r^2

where r is the radius. When graphed, a circle looks like this:


circle


In this circle, the radius is equal to the square root of 25, which is 5.

Ellipses

A conic section that looks very similar to a circle is the ellipse. An ellipse looks like a circle that has been flattened in one direction and stretched out in the other. To get an ellipse from a cone, you would cut completely through the cone, but at an angle that is something other than a right angle.

Instead of having a radius, like a circle, an ellipse has two axes because it's longer in one direction than the other. The longer axis is called the semi-major axis, and the shorter one is the semi-minor axis.

In the equation describing an ellipse, there are two important numbers, designated a and b. a is the length of the semi-major axis, and b is the length of the shorter, semi-minor axis, so a is always larger than b.

(x^2/ a^2) + (y^2/ b^2) = 1 (is the equation that describes the semi-major axis in the x direction)

(x^2/ b^2) + (y^2/ a^2) = 1 (is the equation that describes the semi-major axis in the y direction)

In this ellipse, a = 3 and b = 2, and you can see that these are the lengths of the semi-major and semi-minor axes.


ellipse - conic sections


Ellipses occur in nature quite commonly. For example, as each planet in our solar system orbits the sun, it follows an elliptical path. Yep, even the earth's orbit is an ellipse!

Parabolas

A parabola is formed by slicing through the cone so that a piece is removed from one side, but the cone is not completely cut in half. This creates what is known as an open curve because it is not closed in like a circle or an ellipse, but opens up on one side.

Mathematically, a parabola is defined as the set of points that are the same distance from a given line, called the directrix, and a point, called the focus.

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