Question

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

• A. circle
• B. Diameter of $S_1$
• C. Arc of a circle
• D. chord of $S_1$ but not diameter

Answer 1

Answer: (C)

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