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Conic Sections

Intersections of a plane and double cone creating circles, ellipses, parabolas, and hyperbolas.

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Conic Sections

Conic sections are a family of graphs that include circles and parabolas. Why are they called conic sections

Conic Sections

The conic sections are the shapes that can be created when a plane intersects a double cone like the one below. In other words, the conic sections are the cross sections of a double cone.

There are four primary conic sections - the circle, the parabola, the ellipse, and the hyperbola. These conic sections are shown below with their general equations. In geometry, you will focus on circles and parabolas. You will study ellipses and hyperbolas in more depth in future courses.

 Conic Section Equation Picture Circle (x−h)2+(y−k)2=r2\begin{align*}(x-h)^2+(y-k)^2=r^2 \end{align*} Parabola (x−h)2=4p(y−k)\begin{align*}(x-h)^2=4p(y-k) \end{align*} OR (y−k)2=4p(x−h)\begin{align*}(y-k)^2=4p(x-h)\end{align*} Ellipse (x−h)2a2+(y−k)2b2=1\begin{align*}\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1\end{align*} Hyperbola (x−h)2a2−(y−k)2b2=1\begin{align*}\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2}=1\end{align*} OR (y−k)2a2−(x−h)2b2=1\begin{align*}\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} =1\end{align*}

Let's look at some problems involving conic sections.

1. How is a circle created as the intersection of a double cone and a plane?

The cross section is taken parallel to the base of the cones, as shown below.

2. How is an ellipse created as the intersection of a double cone and a plane?

The intersection plane is at a slight angle with the plane containing the base, as shown below.

3. a) (x1)24+(y+3)29=1\begin{align*}\frac{(x-1)^2}{4}+\frac{(y+3)^2}{9}=1\end{align*}

Ellipse

b) (x4)2=4(y1)\begin{align*}(x-4)^2=4(y-1)\end{align*}

Parabola

c) (x2)2+(y3)2=16\begin{align*}(x-2)^2+(y-3)^2=16\end{align*}

Circle

d) (x1)24(y+3)29=1\begin{align*}\frac{(x-1)^2}{4} - \frac{(y+3)^2}{9}=1\end{align*}

Hyperbola

Notice that the only equation where both the x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} terms are not squared is the parabola. The difference between the ellipse and hyperbola equations is with an ellipse the coefficients of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} are the same sign while with a hyperbola the coefficients of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} are different signs. The coefficients of the x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} terms are always exactly the same for a circle.

Examples

Example 1

Earlier, you were asked why conic sections are named accordingly.

Conic sections are called conic sections because they are the cross sections of a double cone. The word conic comes from the word cone, and the word section comes from cross section.

Example 2

How is a parabola created as the intersection of a double cone and a plane?

The intersection plane has the same slope as the lateral surface of the cone and passes through just one cone. Below, the same intersection is shown from two different perspectives.

Example 3

How is an ellipse created as the intersection of a double cone and a plane?

The intersection plane is at an angle with the plane containing the base such that it passes through both cones, as shown below.

Example 4

Are there any other cross sections of a double cone besides the four primary conic sections?

There are three other cross sections -- a point, a line, and a pair of intersecting lines. These are referred to as the degenerate conics, because they are not as complex. The study of conic sections focuses on the four primary conic sections and not on the degenerate conics.

Review

1. What are the four primary conic sections?

2. Why are the conic sections called conic sections?

Name each of the conic sections based on its picture or equation.

3. (x2)2+(y+5)2=1\begin{align*}(x-2)^2+(y+5)^2=1\end{align*}

4.

5. (x2)24+(y+5)29=1\begin{align*}\frac{(x-2)^2}{4} + \frac{(y+5)^2}{9}=1\end{align*}

6. (x2)24+(y+5)29=1\begin{align*}- \frac{(x-2)^2}{4} + \frac{(y+5)^2}{9}=1\end{align*}

7.

8. (x2)2=4(y5)\begin{align*}(x-2)^2=4(y-5)\end{align*}

9.

10. (y+2)24+(x+5)29=1\begin{align*}\frac{(y+2)^2}{4} + \frac{(x+5)^2}{9}=1\end{align*}

11.

12. x2+y2=9\begin{align*}x^2+y^2=9\end{align*}

13. x=14(y1)2+5\begin{align*}x=\frac{1}{4} (y-1)^2+5\end{align*}

14. What are the degenerate conics?

15. Why are the degenerate conics called degenerate conics?

To see the Review answers, open this PDF file and look for section 10.1.

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