Q. 2 F5.0( 1 Vote )

# Find the equation of the hyperbola whosefocus is (2, 2), directrix is x + y = 9 and eccentricity = 2

Given: Equation of directrix of a hyperbola is x + y – 9 = 0. Focus of hyperbola is (2, 2) and eccentricity (e) = 2

To find: equation of hyperbola

Let M be the point on directrix and P(x, y) be any point of hyperbola

Formula used:

where e is eccentricity, PM is perpendicular from any point P on hyperbola to the directrix

Therefore,

Squaring both sides:

{ (a – b)2 = a2 + b2 + 2ab &

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac}

x2 + 4 – 4x + y2 + 4 – 4y = 2{x2 + y2 + 81 + 2xy – 18y – 18x}

x2 – 4x + y2 + 8 – 4y = 2x2 + 2y2 + 162 + 4xy – 36y – 36x

x2 – 4x + y2 + 8 – 4y – 2x2 – 2y2 – 162 – 4xy + 36y + 36x = 0

– x2 – y2 + 32x + 32y + 4xy – 154 = 0

x2 + y2 – 32x – 32y + 4xy + 154 = 0

This is the required equation of hyperbola.

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