Let P 3 , 3 be a point on the hyperbola, x 2 a 2 - y 2 b 2 = 1 . If the normal to it at P intersects the x -axis at 9 , 0 and e is its eccentricity, then the ordered pair a 2 , e 2 is equal to:

Step-by-step solution

Given hyperbola is x 2 a 2 − y 2 b 2 = 1 . P 3 , 3 lies on hyperbola then 1 a 2 − 1 b 2 = 1 9     . . . 1 Normal at 3 ,   3 is a 2 x 3 + b 2 y 3 = a 2 + b 2 Normal passes through 9 ,   0 3 a 2 = a 2 + b 2   ⇒ 2 a 2 = b 2 Then using ( 1 ) , 1 a 2 − 1 2 a 2 = 1 9 2 a 2 = 9 ⇒ a 2 = 9 2 and b 2 = 9 e 2 = 1 + b 2 a 2 = 1 + 2 = 3 .