The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola x ⁡ 2 4 - y ⁡ 2 5 = 1 , meets the x -axis and y -axis at A and B , respectively. Then O A 2 - O B 2 , where O is the origin, equals

Step-by-step solution

Hyperbola, x 2 a 2 - y 2 b 2 = 1   ⇒ x 2 4 - y 2 5 = 1 ∴  a 2 = 4 ,   b 2 = 5 b 2 = a 2 e 2 - 1   ⇒ 5 = 4 e 2 - 1 ⇒ e 2 = 5 4 + 1 = 9 4 ⇒ e = 3 2 ∴ Extremity of L R in first quadrant ≡ L ≡ a e , b 2 a ≡ 3 ,  5 2 Equation of tangent to the hyperbola x 2 a 2 - y 2 b 2 = 1 at x 1 ,   y 1 is x ⁡ x ⁡ 1 a 2 - y ⁡ y ⁡ 1 b 2 = 1 ⇒ x ⁡ · 3 4 - y · 5 2 5 = 1 ⇒ 3 x ⁡ 4 - y ⁡