Let the centre of a circle, passing through the points (0,0),(1,0) and touching the circle…

Question

Let the centre of a circle, passing through the points $(0,0),(1,0)$ and touching the circle $x^2+y^2=9$, be $(h, k)$. Then for all possible values of the coordinates of the centre $$(h, k), 4\left(h^2+k^2\right)$$ is equal to \_\_\_\_\_\_\_...

Answer: 9

Accepted Answer

Circle will touch internally

$\begin{gathered} {C}_{1}{C}_{2}=|{r}_{1}-{r}_{2}| \ =\sqrt{{h}^{2}+{k}^{2}}=3-\sqrt{{h}^{2}+{k}^{2}} \ \Rightarrow 2\sqrt{{h}^{2}+{k}^{2}}=3 \ \Rightarrow {h}^{2}+{k}^{2}=\frac{9}{4} \ 	herefore 4({h}^{2}+{k}^{2})=9 \end{gathered}$

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