Let P be a point on the hyperbola H : x 2 9 - y 2 4 = 1 , in the first quadrant such that the area of triangle formed by P and the two foci of H is 2 13 . Then, the square of the distance of P from the origin is

Step-by-step solution

Plotting the diagram of given data we get, Given equation of hyperbola is x 2 9 - y 2 4 = 1 . ⇒ a 2 = 9 , b 2 = 4 We know that, b 2 = a 2 e 2 - 1 ⇒ e 2 = 1 + b 2 a 2 ⇒ e 2 = 1 + 4 9 ⇒ e 2 = 13 9 ⇒ e = 13 3 Now, S 1 S 2 = 2 a e ⇒ S 1 S 2 = 2 × 3 × 13 3 ⇒ S 1 S 2 = 2 13 It is given that, area of Δ P S 1 S 2 is 2 13 . ⇒ 1 2 × β × S 1 S 2 = 2 13 ⇒ 1 2 × β × 2 13 = 2 13 ⇒ β = 2 Since, α , β lies on the hyperbola, ⇒ α 2 9 - β 2 4 = 1 ⇒ α 2 9 - 1 = 1 ⇒ α 2 = 18 ⇒ α = 3 2 Distance of P from origin is given by, O P = α 2 +