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Q.
The locus of the mid points of the chords of the circle $x^2+y^2=16$, which are tangent to the hyperbola $9 x^2-16 y^2=144$ is
Conic Sections
Solution:
Let $(h, k)$ be the mid point of the chord of the circle $x^2+y^2=16$, so that its equation by $T = S _1$ is $hx + ky = h ^2+ k ^2$
or $y =-\frac{ h }{ k } x +\frac{ h ^2+ k ^2}{ k }$
i.e. the form $y = mx + c$
It will touch the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$
if $c ^2=16 m ^2-9$
$\therefore \quad\left(\frac{ h ^2+ k ^2}{ k }\right)^2=16\left(-\frac{ h }{ k }\right)^2-9 $
$\left( h ^2+ k ^2\right)^2=16 h ^2-9 k ^2$
Generalising, the locus of the mid-point $(h, k)$ is $\left(x^2+y^2\right)^2=16 x^2-9 y^2$