A line parallel to the straight line 2 x - y = 0 is tangent to the hyperbola x 2 4 − y 2 2 = 1 at the point x 1 , y 1 . Then x 1 2 + 5 y 1 2 is equal to

Step-by-step solution

Tangent at x 1 , y 1 x x 1 - 2 y y 1 - 4 = 0 This is parallel to 2 x - y = 0 ⇒    x 1 2 y 1 = 2 ⇒    x 1 = 4 y 1     . . . 1 Point x 1 , y 1 lie on hyperbola x 1 2 4 - y 1 2 2 - 1 = 0     . . . 2 Using equation 1 and 2 , 4 y 1 2 - y 1 2 2 = 1 ⇒ y 1 2 = 2 7   &   x 1 2 = 32 7 We get x 1 2 + 5 y 1 2 = 6