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  4. Two circles are S1 ≡(x+3)2+y2=9 S2 ≡(x-5)2+y2=16 with centres C1 C2 From point ' A ' on S2 which is nearest to S1, a variable chord is drawn to S1. The locus of mid point of the chord is

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Q. Two circles are
$S_1 \equiv(x+3)^2+y^2=9 $
$ S_2 \equiv(x-5)^2+y^2=16$
with centres $C_1 \& C_2$
From point ' $A$ ' on $S_2$ which is nearest to $S_1$, a variable chord is drawn to $S_1$. The locus of mid point of the chord is

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Solution:

Let mid point be $N(h, k)$. Now equation of chord $L M$ is $T=S_1$ $\Rightarrow hx + ky +3( x + h )= h ^2+ k ^2+6 h$
As it passes through $(1,0) \Rightarrow h+3(1+h)=h^2+k^2+6 h$
So locus is $x^2+y^2+2 x-3=0$ which is a circle with centre $(-1,0)$ and radius 2 . But it is clear from geometry that it will be major arc $BC _1 C$ as shown in figure.
image