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 ZZ^{1} 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.

I) Find the equation of the hyperbola whose

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

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