Q. 75.0( 3 Votes )

If the latus-rect

Answer :

Given: latus-rectum through one focus of a hyperbola subtends a right angle at the farther vertex


To find: eccentricity of hyperbola


Let B is vertex of hyperbola and A and C are point of intersection of latus-rectum and hyperbola



Standard equation of hyperbola is



Since, A and C lie on hyperbola


Therefore







As angle between AB and AC is 90°


SlopeAB × SlopeBC = -1





From (i):




{ b2 = a2(e2 – 1)}



(e2 – 1)2 = (e + 1)2


e4 + 1 – 2e2 = e2 + 1 + 2e


e4 + 1 – 2e2 – e2 – 1 – 2e = 0


e4 – 3e2 – 2e = 0


e(e3 – 3e – 2) = 0


e(e– 2)(e + 1)2 = 0


e = 0 or 2 or -1


But e should be greater than or equal to 1 for hyperbola


e = 2


Hence, eccentricity of hyperbola is 2


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