# If the latus-rect

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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