Consider a circle , where . If the circle touches the line y + x = 0 at the point P, whose distance from the origin is , then is equal to __________. [2024]

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Solution

Q.
Consider a circle ${(x - α)}^{2} + {(y - β)}^{2} = 50$ , where $α, β > 0$ . If the circle touches the line y + x = 0 at the point P, whose distance from the origin is $4 \sqrt{2}$ , then ${(α + β)}^{2}$ is equal to __________. [2024]

Ans.
(100)

We have circle,

${(x - α)}^{2} + {(y - β)}^{2} = 50$

Centre of (i), is $C (α, β)$ and radius, $r = 5 \sqrt{2}$

$∴ C P = r$

$\Rightarrow$ $| \frac{α + β}{\sqrt{2}} |$ $= 5 \sqrt{2}$

$\Rightarrow | α + β | = 10 \Rightarrow {(α + β)}^{2} = 100$ .

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