Find the equation of the hyperbola whose

focus is (1, 1), directrix is 3x + 4y + 8 = 0 and eccentricity = 2

Given: Equation of directrix of a hyperbola is 3x + 4y + 8 = 0. Focus of hyperbola is (1, 1) 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}


⇒ 25{x2 + 1 – 2x + y2 + 1 – 2y} = 4{9x2 + 16y2+ 64 + 24xy + 64y + 48x}


⇒ 25x2 + 25 – 50x + 25y2 + 25 – 50y = 36x2 + 64y2 + 256 + 96xy + 256y + 192x


⇒ 25x2 + 25 – 50x + 25y2 + 25 – 50y – 36x2 – 64y2 – 256 – 96xy – 256y – 192x = 0


⇒ – 11x2 – 39y2 – 242x – 306y – 96xy – 206 = 0


⇒ 11x2 + 39y2 + 242x + 306y + 96xy + 206 = 0


This is the required equation of hyperbola



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