If the focal chord of the hyperbola subtends a right angle at the center, then its eccentricity is

Step-by-step solution

DTP REQUIRED... eccentricity   of   hyperbola = 1 + b 2 a 2 Distance A B = x 2 - x 1 2 + y 2 - y 1 2 ⇒ a e - a e 2 + - b 2 a - b 2 a 2 ⇒ 4 b 4 a 2 = 2 b 2 a Now, tan   45 ° = b 2 a 1 = b 2 a a e ⇒ e = b 2 a 2 Now, e = 1 + b 2 a 2 e 2 = 1 + b 2 a 2 ⇒ b 2 a 2 = e 2 - 1 e = e 2 - 1 ⇒ e 2 - e - 1 = 0 Which is quadratic equation compare it with a x 2 + b x + c We get a = 1 , b = - 1 , c = - 1 e = - 1 ± b 2 - 4 a c 2 a = - - 1 ± 1 2 - 4 1 - 1 2 × 1 e = 1 &#