## Ellipses

#### Definition of an ellipse

An ellipse is the locus of a point that moves in the plane in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed straight line is always constant and is always less than unity.

The constant ratio is denoted by e and is known as the eccentricity of the ellipse.

If S is the focus, ZZ1 is the directrix and P is any point on the ellipse, then the definition

SP/PM = e

#### Example 1

Find the equation of the ellipse whose focus is (1, 0) and directrix is x + y + 1 = 0.

Let S(1, 0) be the focus, ZZ1 be the directrix and the point P(x,y) be any point on the ellipse.

PM = perpendicular from P onto the directrix then

SP/PM = e

#### Try these questions

1.    Find the equation of the ellipse where

 a. Focus is (-1, 1) Directrix x - 2y + 3= 0 and e = 1/3 b. Focus is (-2, 3) Directrix is 2x + 3y + 4 = 0, e = 4/5 c. Focus is (1, 2) Directrix is 3x + 4y - 5 = 0, e = 1/2

#### Answers

a.     Solution

b.     Solution

Given that the focus S = (-2,3)

Directrix is 2x + 3y + 4= 0

e = 4/5

If P(x,y) is a point on the ellipse

PM = perpendicular from P onto the directrix

Cross-multiplying

c.     Solution

Focus = (1, 2) = S

Directrix is 3x + 4y - 5 = 0

Eccentricity e = 1/2

If P(x,y) be a point on the ellipse

PM = perpendicular from P onto the directrix