If the tangents drawn to the hyperbola 4 y^2=x^2+ 1 intersect the co-ordinate axes at the distinct points A and B , then the locus of the mid point of A B is

Step-by-step solution

Equation of hyperbola is : aligned & 4 y^2=x^2+1 -x^2+4 y^2=1 &- x^2 1^2 + y^2 ( 1 2 )^2 =1 & a=1, b= 1 2 aligned Now, tangent to the curve at point (x₁, y₁ ) is given by. aligned &4 2 y₁ d y d x =2 x₁ & d y d x = 2 x₁ 8 y₁ = x₁ 4 y₁ aligned Equation of tangent at (x₁, y₁ ) is aligned &y=m x+c & y= x₁ 4 y₁ x+c aligned As tangent passes through x₁, y₁ aligned & y₁= x₁ x₁ 4 y₁ +c & C= 4 y₁^2-x₁^2 4 y₁ = 1 4 y₁ aligned Therefore, y= x₁ 4 y₁ x+ 1 4 y₁ 4 y₁ y=x₁ x+1 which intersects x axis at A ( -1 x₁ , 0 ) and y axis