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 :
Correct Option: , 4
Solution:
$y-k=-\frac{h}{k}(x-h)$
$\mathrm{ky}-\mathrm{k}^{2}=-\mathrm{hx}+\mathrm{h}^{2}$
$\mathrm{hx}+\mathrm{ky}=\mathrm{h}^{2}+\mathrm{k}^{2}$
$\mathrm{y}=-\frac{\mathrm{hx}}{\mathrm{k}}+\frac{\mathrm{h}^{2}+\mathrm{k}^{2}}{\mathrm{k}}$
tangent to $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$
$\mathrm{c}^{2}=\mathrm{a}^{2} \mathrm{~m}^{2}-\mathrm{b}^{2}$
$\left(\frac{\mathrm{h}^{2}+\mathrm{k}^{2}}{\mathrm{k}}\right)^{2}=9\left(-\frac{\mathrm{h}}{\mathrm{k}}\right)^{2}-16$
$\left(x^{2}+y^{2}\right)^{2}=9 x^{2}-16 y^{2}$