Circle 2 Question 21
22. Let $C _1$ and $C _2$ be two circles with $C _2$ lying inside $C _1$. A circle $C$ lying inside $C _1$ touches $C _1$ internally and $C _2$ externally. Identify the locus of the centre of $C$.
$(2001,5 M)$
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Solution:
- Let the given circles $C _1$ and $C _2$ have centres $O _1$ and $O _2$ and radii $r _1$ and $r _2$, respectively.
Let the variable circle $C$ touching $C _1$ internally, $C _2$ externally have a radius $r$ and centre at $O$.
Now, $\quad O O _2=r+r _2$ and $O O _1=r _1-r$
$\Rightarrow \quad O O _1+O O _2=r _1+r _2$
which is greater than $O _1 O _2$ as $O _1 O _2<r _1+r _2$
$\left[\because C _2\right.$ lies inside $\left.C _1\right]$
$\Rightarrow$ Locus of $O$ is an ellipse with foci $O _1$ and $O _2$.
Alternate Solution
Let equations of $C _1$ be $x^{2}+y^{2}=r _1^{2}$ and of $C _2$ be $(x-a)^{2}+(y-b)^{2}=r _2^{2}$
Let cetnre $C$ be $(h, k)$ and radius $r$, then by the given condition
$$ \begin{aligned} \sqrt{(h-a)^{2}+(k-b)^{2}}=r+r _2 \text { and } \sqrt{h^{2}+k^{2}} & =r _1-r \\ \Rightarrow \quad \sqrt{(h-a)^{2}+(k-b)^{2}}+\sqrt{h^{2}+k^{2}} & =r _1+r _2 \end{aligned} $$
Required locus is
$$ \sqrt{(x-a)^{2}+\left(y-b^{2}\right)}+\sqrt{x^{2}+y^{2}}=r _1+r _2 $$
which represents an ellipse whose foci are $(a, b)$ and $(0,0)$.
