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## 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

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