Question

Let a variable point $P$ moves on the circle $x^2+y^2=9$ and the line $x+y=3$ cuts the co-ordinates axes at $A$ and $B$. Also locus of the centroid of $\triangle P A B$ is the curve $C$. If the locus of the centre of a variable circle which touches the circle $x^2+y^2=9$ and curve $C$ is an ellipse whose eccentricity is e, find the value of $e ^{-2}$.

Answer 1

$h=\frac{3 \cos \theta+0+3}{3}=\cos \theta+1 $
$\therefore k=\frac{3 \sin \theta+3+0}{3}=\sin \theta+1 $
$\therefore(h-1)^2+(k-1)^2=1$
$(x-1)^2+(y-1)^2=1 $
$x^2-2 x+y^2-2 y+1=0$
$C: x^2+y^2-2 x-2 y+1=0$

$CC _1= r + r _1 $
$CC _2= r _2- r $
$ CC _1+ CC _2= r _1+ r _2=1+3=4=2 a$
Hence locus of $C$ is an ellipse.
Now $2 a =4$ and $2 ae =\sqrt{2} \Rightarrow e =\frac{\sqrt{2}}{4}=\frac{1}{2 \sqrt{2}}$
Hence $\frac{1}{ e ^2}=8$