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Q.
Locus of the centre of the circle which touches the two circles $x^2+y^2+8 x-9=0$ and $x^2+y^2-8 x+7=0$ externally, is a conic whose
Conic Sections
Solution:
As, $\left| cc _1- cc _2\right|=\left|\left( r + r _1\right)-\left( r + r _2\right)\right|=$ constant
where $\left| r _1- r _2\right|< c _1 c _2$
$\Rightarrow$ locus of $C$ is a hyperbola with foci $c _1$ and $c _2$ i.e., $(-4,0)$ and $(4,0)$.
Also, $2 a =\left| r _1- r _2\right|=2 \Rightarrow a =1$
Now, $e =\frac{2 ae }{2 a }=\frac{8}{2}=4$
So, $b^2=1^2\left(4^2-1\right)=15$
Hence, locus of centre of circle is hyperbola, whose equation is $\frac{x^2}{1}-\frac{y^2}{15}=1$.
