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  4. The locus of centre of a variable circle touching the circles of radius r1 and r2 externally, which also touch each other externally, is a conic of the eccentricity e. If (r1/r2)=3+2 √2, then e2 is

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Q. The locus of centre of a variable circle touching the circles of radius $r_1$ and $r_2$ externally, which also touch each other externally, is a conic of the eccentricity e. If $\frac{r_1}{r_2}=3+2 \sqrt{2}$, then $e^2$ is

Conic Sections

Solution:

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$cc _2= r + r _2 $
$cc _1- cc _2= r _1- r _2$
$\Rightarrow$ Locus of $c$ is hyperbola with foci $c _1$ and $c _2$
$\Rightarrow 2 ae = c _1 c _2= r _1+ r _2$ and $r _1- r _2=2 a$
$\Rightarrow \frac{ r _1+ r _2}{ r _1- r _2}=\sqrt{2}$
$\Rightarrow e ^2=2 $