Q. 2 F5.0( 1 Vote )

Find the equation of the hyperbola whose

focus is (2, 2), directrix is x + y = 9 and eccentricity = 2

Answer :

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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