The locus of the midpoints of the chord of the circle, x^2+y^2=25 which is tangent to the hyperbola

The locus of the midpoints of the chord of the circle, x^2+y^2=25 which is tangent to the hyperbola, x^2/9-y^2/16=1 is [Main 16th March 1st Shift 2021]
(a) (x^2+y^2 )^2-9x^2-16y^2=0
(b) (x^2+y^2 )^2-9x^2+144y^2=0
(c) (x^2+y^2 )^2-16x^2+9y^2=0
(d) (x^2+y^2 )^2-9x^2+16y^2=0
Ans: d
Sol.
Let (h,k) be the mid-point of the chord of circle x^2+y^2=25 with centre (0, 0).
∴ Equation of chord is hx+ky=h^2+k^2
⇒y=-h/k x+(h^2+k^2)/k ...(i)
Now, (i) will be tangent to hyperbola
x^2/9-y^2/16=1 if c^2=a^2 m^2-b^2
⇒((h^2+k^2)/k)^2=9((-h)/k)^2-(16) [Using (i)]
⇒(h^2+k^2 )^2=9h^2-16k^2
∴ Required locus is (x^2+y^2 )^2-9x^2+16y^2=0



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