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:
Equation of chord
$y-k=-\frac{h}{k}(x-h)$
$k y-k^{2}=-h x+h^{2}$
$h x+k y=h^{2}+k^{2}$
$y=-\frac{h x}{k}+\frac{h^{2}+k^{2}}{k}$
tangent to $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$
$c^{2}=a^{2} m^{2}-b^{2}$
$\left(\frac{h^{2}+k^{2}}{k}\right)^{2}=9\left(-\frac{h}{k}\right)^{2}-16$
$\left(x^{2}+y^{2}\right)^{2}=9 x^{2}-16 y^{2}$