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°

⇒ Slope_AB × Slope_BC = -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