Let the eccentricity of the hyperbola H : x 2 a 2 - y 2 b 2 = 1 be 5 2 and length of its latus rectum be 6 2 . If y = 2 x + c is a tangent to the hyperbola H , then the value of c 2 is equal to

Step-by-step solution

Given equation of the hyperbola is x 2 a 2 - y 2 b 2 = 1 If e is eccentricity of the hyperbola, then e 2 = 1 + b 2 a 2 1 + b 2 a 2 = 5 2 ⇒ b 2 = 3 a 2 2 Length of the latus-rectum is given by 2 b 2 a i.e. 2 b 2 a = 6 2 2 a × 3 a 2 2 = 6 2 ⇒ a = 2 2 and b 2 = 3 2 × 8 = 12 ⇒ b = 2 3 If y = m x + c is a tangent to the hyperbola, then c 2 = a 2 m 2 - b 2 ∴   c 2 = 4 × 8 - 12 Hence c 2 = 20