## Hyperbolas

#### Definition of a hyperbola

A hyperbola 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 (focus) in the same plane to its distance from a fixed line (directrix) in the plane is always constant and greater than unity.

As in the previous cases, the constant ratio is the eccentricity e and e > 1 for every hyperbola.

The fixed point is the focus S, the fixed line is the directrix ZZ1 and if P(x,y) is any point on the hyperbola, then

SP/PM = e

⇒ SP= e.PM

Example 1:

Find the equation of the hyperbola whose focus is (1, 2), directrix is the line x+y+1=0 and eccentricity is e = 3/ 2.

Solution:

Let P(x,y) be a point on the hyperbola.

Let PM = perpendicular from P onto the directrix By cross-multiplying is the equation of the required hyperbola.

#### Try these questions

I)    Find the equation of the hyperbola whose

1. Focus is (-1, 1) Directrix x - y + 3= 0 and eccentricity is 3
2. Focus is (2, -1) Directrix is 2x + 3y = 1 and eccentricity is 2
3. Focus is (a, 0) Directrix is 2x - y + a = 1 and eccentricity is 4/3
4. Focus is (2, 2) Directrix is x + y = 9 and eccentricity is 3/2

i.    Solution:  is the equation of the required hyperbola.

ii.    Solution:

Given focus =S = (2, -1)

Directrix is 2x + 3y = 1 => 2x + 3y - 1 = 0

e = 2

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix Cross multiplying is the equation of the required hyperbola.

iii.    Solution:

Given focus =S = (a, 0)

Directrix is 2x - y + a = 0

e = 4/3

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix is the equation of the required hyperbola.

iv.    Solution:

Given focus =S= (2, 2)

Directrix is x + y = 9 => x + y - 9 = 0

e = 3/2

Let P(x,y) be any point on the hyperbola

Let PM = perpendicular from P onto the directrix is the equation of the required hyperbola.