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Definition of a circle

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Question

If the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ is touched by $y = x$ at P such that $O P = 6 \sqrt{2},$ then the value of c is

Moderate

Solution

The equation of the line y = x in distance form is $\frac{x}{\cos θ} = \frac{y}{\sin θ} = r, where θ = \frac{π}{4}$
For point $p$ we have $r = 6 \sqrt{2} .$
Therefore, coordinates of Pare given by
$\frac{x}{\cos \frac{π}{4}} = - \frac{y}{\sin \frac{π}{4}} = 6 \sqrt{2} \Rightarrow x = 6, y = 6$
Since P(6, 6) lies on $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
$∴ 72 + 12 (g + f) + c = 0$
Since y = x touches the circle. Therefore, the equation
$2 x^{2} + 2 x (g + f) + c = 0$
$\Rightarrow 4 (g + f)^{2} = 8 c \Rightarrow (g + f)^{2} = 2 c$
$\begin{aligned} [12 (g + f)]^{2} = [- (c + 72)]^{2} \\ \Rightarrow 144 (g + f)^{2} = (c + 72)^{2} \\ \Rightarrow 144 (2 c) = (c + 72)^{2} \\ \Rightarrow (c - 72)^{2} = 0 \Rightarrow c = 72 \end{aligned}$

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