Let the centre of a circle, passing through the points (0, 0), (1, 0) and touching the circle , be (h, k). Then for all possible values of the coordinates of the centre (h, k) is equal to _________. [2024]

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Solution

Q.
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), $4 (h^{2} + k^{2})$ is equal to _________. [2024]

Ans.
(9)

Circle will touch internally

$∴ C_{1} C_{2} = | r_{1} - r_{2} |$

$\Rightarrow \sqrt{h^{2} + k^{2}} = 3 - \sqrt{h^{2} + k^{2}} \Rightarrow 2 \sqrt{h^{2} + k^{2}} = 3$

$\Rightarrow h^{2} + k^{2} = \frac{9}{4} ∴ 4 (h^{2} + k^{2}) = 9$ .

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