Question

The locus of the midpoints of the chord of the circle, $x^2+y^2=25$ which is tangent to the hyperbola, $\frac{x^2}{9}-\frac{y^2}{16}=1$ is:

• A. $(x^2+y^2{)}^2-16x^2+9y^2=0$
• B. $(x^2+y^2{)}^2-9x^2+144y^2=0$
• C. $(x^2+y^2{)}^2-9x^2-16y^2=0$
• D. $(x^2+y^2{)}^2-9x^2+16y^2=0$

Answer 1

The correct option is D $(x^2+y^2{)}^2-9x^2+16y^2=0$

tangent of hyperbola
$y=mx\pm \sqrt{9m^2-16}\cdots \left(i\right)$
which is a chord of circle with mid - point $(h,k)$
so equation of chord $T=S_1$
$hx+ky=h^2+k^{2}y=-\frac{hx}{k}+\frac{h^2+k^2}{k}\cdots \left(ii\right)$
by $\left(i\right)$ and $\left(ii\right)$
$m=\frac{h}{k}\text{and}\pm \sqrt{9m^2-16}=\frac{h^2+k^2}{k}9\frac{h^2}{k^2}-16=\frac{(h^2+k^2{)}^2}{k^2}\text{locus}9x^2-16y^2=(x^2+y^2{)}^2$