Q. 15.0( 6 Votes )

The equation of t

Answer :

Given: Equation of directrix of a hyperbola is x – y + 3 = 0. Focus of hyperbola is (-1, 1) and eccentricity (e) = 3


To find: equation of the hyperbola


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


Formula used:



where e is an 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}


2{x2 + 1 + 2x + y2 + 1 – 2y} = 9{x2 + y2+ 9 + 6x – 6y – 2xy}


2x2 + 2 + 4x + 2y2 + 2 – 4y = 9x2 + 9y2+ 81 + 54x – 54y – 18xy


2x2 + 4 + 4x + 2y2– 4y – 9x2 - 9y2 - 81 – 54x + 54y + 18xy = 0


– 7x2 - 7y2 – 50x + 50y + 18xy – 77 = 0


7x2 + 7y2 + 50x – 50y – 18xy + 77 = 0


This is the required equation of hyperbola



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