Find the equation of the hyperbola whose

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

Given: Equation of directrix of a hyperbola is x + y – 1 = 0. Focus of hyperbola is (0, 3) and eccentricity (e) = 2


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 + y2 + 9 – 6y} = 4{x2 + y2 + 1 – 2x – 2y + 2xy}


⇒ 2x2 + 2y2 + 18 – 12y = 4x2 + 4y2+ 4 – 8x – 8y + 8xy


⇒ 2x2 + 2y2 + 18 – 12y – 4x2 – 4y2 – 4 – 8x + 8y – 8xy = 0


⇒ – 2x2 – 2y2 – 8x – 4y – 8xy + 14 = 0


⇒ -2(x2 + y2 – 4x + 2y + 4xy – 7) = 0


⇒ x2 + y2 – 4x + 2y + 4xy – 7 = 0


This is the required equation of hyperbola



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