9.4 Graph and Write Equations of Ellipses

Ellipse

The set of all points P such that the sum of the distances between P and two fixed points, called the foci, is a constant

Foci

Two fixed points in an ellipse

Vertices

The points at which the line through the foci intersect the ellipse

Major axis

The line segment that joins the vertices

Center

The midpoint of the major axis

Co-vertices

The points of intersection of an ellipse and the line perpendicular to the major axis at the center

Minor axis

The line segment that joins the co-vertices

STANDARD EQUATION OF AN ELLIPSE WITH CENTER AT THE ORIGIN

Equation	Major Axis	Vertices	Co-Vertices
	Horizontal	(± _a_, 0)	(0, ± _b_)
	Vertical	(0, ± _a_)	(± _b_, 0)
The major and minor axes are of lengths 2a and 2b, respectively, where a > b > 0. The foci of the ellipse lie on the major axis at a distance of c units from the center, where c^2 = _a^2 - b^2_

Example 1

Graph the equation 9x² + 36y² = 324. Identify the vertices, co-vertices, and foci of the ellipse.

1. Rewrite the equation in standard form.

9x^2 + 36y^2 = 324	Write original equation.
	Divide each side by _324_.
	Simplify.

2. Identify the vertices, co-vertices, and foci. Note that a² = _36_ and b² = _9_, so a = _6_ and b = _3_. The denominator of the x²-term is _greater than_ that of the y²-term, so the major axis is __horizontal__. The vertices of the ellipse are at (±a, 0) = (± _6_, 0). The co-vertices are at (0, ±b) = (0, ± _3_). Find the foci.

c² = a² - b² = _6² - 3²_ = _27_, so c = .

The foci are at (± , 0), or about (± _5.2_, 0).

3. Draw the ellipse that passes through each vertex and co-vertex.

Example 2

Write an equation of the ellipse that has a vertex at (0, 7), a co-vertex at (-4, 0), and center at (0, 0).

Sketch the ellipse as a check for your final equation. By symmetry, the ellipse must also have a vertex at (0, _-7_) and a co-vertex at (_4_, 0).

Because the vertex is on the _y-axis_ and the co-vertex is on the _x-axis_, the major axis is _vertical_ with a = _7_, and the minor axis is _horizontal_ with b = _4_.

An equation is , or

Example 3

Write an equation given a vertex and a focus

Write an equation of the ellipse that has a vertex at (-6, 0) and a focus at (5, 0).

Solution

Make a sketch of the ellipse. Because the vertex and focus lie on the _x-axis_, the major axis is _horizontal_, with a = _6_ and c = _5_. To find b, use the equation c² = a² - b².

_5²_ = _6²_ - b²

b² = _6²_ - _5²_ = _11_

b =

An equation is = 1 or = 1.