Chords of the circle x²+y²=r² touch the hyperbola x² a² - y² b² =1 . The locus of the mid-points of the chords is

Step-by-step solution

Let P (x₁, y₁ ) be the mid-point of a chord of x²+y²=r² . Its equation is aligned S₁ &=S₁₁ x₁ x+y₁ y=x₁²+y₁² or y &= (- x₁ y₁ ) x+ x₁²+y₁² y₁ aligned The line y=m x+c is a tangent to the hyperbola only if aligned c² &=a² m²-b² (x₁²+y₁² )² y₁² &= a² x₁² y₁² -b² aligned The locus of P is (x²+y² )²=a² x²-b² y²