# Hyperbola Equation

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Let S be a fixed point and L, a fixed straight line on a plane. If a point P moves on this plane in such a way that its distance from the fixed point S always bears a constant ratio to its perpendicular distance from the fixed line L and if this ratio is greater than unity then the locus traced out by P is called a hyperbola.

Equation of a hyperbola whose center is at origin (0, 0) is

x^2 / a^2 – y^2 / b^2 = 1                … (1)

Question 1: - Find the lengths of axes of the parabola 9 x^2 – 25 y ^2 = 225.

Solution: - 9 x^2 – 25 y ^2 = 225.

x^2 / 25 – y^2 / 9 = 1             … (2)

Comparing equation (2) with the standard form of hyperbola (1) we get,

A^2 = 25 or, a = 5     and b^2 = 9 or, b = 3

Therefore, the length of the transverse axis of the hyperbola (2) is 2 a = 2 * 5= 10

And the length of the conjugate axis = 2 b = 2 * 3 = 6.

Question 2: - If length of the transverse and conjugate axes of a hyperbola is 8 and 12 respectively, then find the equation of the hyperbola.

Solution: - According to the problem,

2 a = 8,          therefor a = 4
And      2 b = 12,         therefor b = 6.

Substituting these values in equation (1) we get,

x^2 / 4^2 – y^2 / 6^2 = 1

x^2/16 – y^2/ 36 = 1

9x^2 – 4y^2 =144