The locus of the mid-points of the chords of the circle x^2+y^2=16 which are the tangents to the hyperbola 9 x^2-16 y^2=144 is

Step-by-step solution

Equation of chord with mid-point (h, k) to the circle x^2+y^2=16 is T=S₁ gathered h x+k y-16=h^2+k^2-16 y=- h k x+ ( h^2+k^2 k ) gathered For hyperbola 9 x^2-16 y^2=144 x^2 16 - y^2 9 =1 As we know that, condition of tangency of a line y=m x+c is a^2 m^2-b^2=c^2 So, aligned So, & & 16 (- h k )^2-9 & = ( h^2+k^2 k )^2 & & 16 h^2-9 k^2 & = (h^2+k^2 )^2 aligned Replace h, k by x, y 16 x^2-9 y^2= (x^2+y^2 )^2