From a point P , two tangents P A and P B are drawn to the hyperbola x 2 a 2 - y 2 b 2 = 1 . If the product of the slopes of these tangents is 1 , then the locus of P is a conic whose eccentricity is equal to

Step-by-step solution

Let, P be h , k and the equation of tangent through P on the hyperbola is y = m x ± a 2 m 2 - b 2 ⇒ k 2 - m h 2 = a 2 m 2 - b 2 ⇒ h 2 - a 2 m 2 - 2 k h m + k 2 + b 2 = 0 where roots are m 1 and m 2 ⇒ k 2 + b 2 h 2 - a 2 = 1 = m 1 m 2 ⇒ locus is x 2 - y 2 = a 2 + b 2 ⇒ eccentricity = 1 + 1 = 2