If the tangent drawn to the hyperbola 4 y 2 = x 2 + 1 intersect the co-ordinates axes at the distinct points A and B , then the locus of the midpoint of A B is :

Step-by-step solution

Let tangent drawn at point x 1 ,   y 1 to the hyperbola 4 y 2 = x 2 + 1 is 4 y y 1 = x x 1 + 1 . This tangent intersect coordinate axes at A and B respectively, then A - 1 x 1 ,   0 &   B 0 ,   1 4 y 1  . Let mid point is M h ,   k . 2 h = - 1 x 1 ⇒ x 1 = - 1 2 h           . . . 1 2 k = 1 4 y 1 ⇒ y 1 = 1 8 k       . . . 2 Since, point x 1 ,   y 1 , lies on the hyperbola. So, 4 y 1 2 = x 1 2 + 1 From equations 1 and 2 , we get