1CEllipses


 * Ellipses**

Ellipses are one of the group of shapes called Conic Sections (not in course). Ellipses can be defined geometrically (not in course). Ellipses have applications in Physics (not in course).

An ellipse is a circle that has been dilated in one direction.

An ellipse centered on the origin can be defined using coordinate geometry as the relation: math \dfrac {x^2}{a^2} + \dfrac{y^2}{b^2} = 1 math

Where a is the radius in the x-direction and b is the radius in the y-direction. (a and b are lengths so must be positive)

Domain: x Î [–a, a] Range: y Î [–b, b]

x-intercepts x = –a, x = a y-intercepts y = –b, y = b

The largest diameter (the larger out of 2a or 2b) is called the __major axis__. The smallest diameter (the smaller out of 2a or 2b) is called the __minor axis__.

Notice that if a = b the equation of an ellipse reduces to the equation of a circle.

General Equation for an Ellipse

If the ellipse is translated h units to the right and k units up, we get the general equation for an ellipse: math \dfrac {(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1 math

Domain: x Î [h – a, h + a] Range: y Î [k – b, k + b]

Example 1

math \dfrac {(x-4)^2}{4} + \dfrac{(y-1)^2}{9} = 1 math

This is an ellipse with centered at (4, 1) with a=2, b=3

Domain: x Î [2, 6] Range: y Î [–2, 4]

Length of major axis = 6 Length of minor axis = 4

Vertices (2, 1) (6, 1) (4, –2) (4, 6)

There are no y-intecepts

x-intercepts (find by setting y=0) math x = 4 \pm \dfrac {4\sqrt{2}}{3} math

Example 2 Sketch the graph of 5x 2 + 9y 2 – 36y – 9 = 0

Aim: To rewrite this equation in the form of the general equation for an ellipse.

5x 2 + 9y 2 – 36y = 9

The x 2 term does not need work since there is no term with x in it.

Take out 9 as a common factor from the y terms.

5x 2 + 9(y 2 – 4y) = 9

Complete the square for the y terms by adding 4 inside the bracket – equivalent to 36 outside the bracket.

5x 2 + 9(y 2 – 4y + 4) = 9 + 36

5x 2 + 9(y – 2) 2 = 45

Divide both sides by 45 math \\ \dfrac {5x^2}{45} + \dfrac {9(y - 2)^2}{45} = \dfrac {45}{45} \\ \\ \dfrac {x^2}{9} + \dfrac {(y - 2)^2}{5} = 1 math

This is an ellipse with a center at (0, 2) and math \\ a = 3, \: b = \sqrt{5} \\ \\ \text {Domain: } x \in \big[ 0 - 3, 0 + 3 \big] \Longrightarrow x \in \big[ -3, 3 \big] \\ \text {Range: } \: y \in \left[ 2 - \sqrt{5}, 2 + \sqrt{5} \right] math



Sketching Ellipses on a Calculator

The best way to draw ellipses is to use parametric mode and the parametric equations of an ellipse.

Return to summary of Conic Sections .