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Definitions:

1. An ellipse is the figure consisting of all those points for which the sum of their distances to two fixed points (called the foci) is a constant.

2. An ellipse is the figure consisting of all points in the plane whose Cartesian coordinates satisfy the equation

Where h, k, a and b are real numbers. a and b are positive.

Formulas:

Equations

An ellipse centered at the point (h, k) and having its major axis parallel to the x-axis may be specified by the equation

Parametric equations of the ellipse:

Major axis = 2a

Minor axis = 2b

Eccentricity

Define a new constant called the eccentricity ( is the case of a circle) The eccentricity is:

.

The greater the eccentricity is, the more elongated is the ellipse.

Foci:

If c equals the distance from the center to either focus, then

The distance between the foci is

Area

The area enclosed by an ellipse is , where In the case of a circle where a = b, the expression reduces to the familiar

Tangent line

Say is a fixed point of the ellipse.

The equation of the tangent line in point of an ellipse

Example 1:

Given the following equation

a Find the length of the major and minor axes.

b) Find the coordinates of the foci.

c) Sketch the graph of the equation.

Solution:

a) We first write the given equation in standard form:

The major axis length is given by = 2a = 4

The minor axis length is given by = 2b = 6

b)

c)

Example 2:

Sketch the graph of the ellipse whose equation is

Solution:

We see that the center of the ellipse is (h, k) = (2, -1). Next, note that a = 3, b = 2.

We know that the endpoints of the major axis are exactly 3 units left and right the center, which places them at the points (-1, -1) and (5, -1).

We also know that the endpoints of the minor axis are exactly 2 units above and below the center, which places them at the points (2, 1)and (2, -3).

Foci: