If the circle $x^2+y^2+2 g x+2 f y+c=0(c0)$ touches both the coordinate axes and lies in the third quadrant, then the length of the chord intercepted by the circle on the line $x+y+\sqrt{c}=0$ is

Question

If the circle $x^2+y^2+2 g x+2 f y+c=0(c>0)$ touches both the coordinate axes and lies in the third quadrant, then the length of the chord intercepted by the circle on the line $x+y+\sqrt{c}=0$ is

Aicharya · Accepted Answer

Given equation of circle $ x^2+y^2+2 g x+2 f y+c=0 \quad(c>0) $ Coordinate of centre $=(-g,-f)$ $ \text { radius }=\sqrt{g^2+f^2-c} $

Circle touch both the axes, so $ \begin{aligned} g^2 & =f^2=c \Rightarrow g= \pm \sqrt{c} \\ f & = \pm \sqrt{c} \end{aligned} $ So, coordinate of centre lies in third quadrant so, centre is $(-\sqrt{c},-\sqrt{c})$ equation of given line $x+y+\sqrt{c}=0$ So, line pass through the point of contact to axes of the circle. $ \text { Hence, intercept } \begin{aligned} A B & =\sqrt{(-\sqrt{c}-0)^2+(0+\sqrt{c})^2} \\ & =\sqrt{c+c}=\sqrt{2 c} \end{aligned} $

If the circle $x^2+y^2+2 g x+2 f y+c=0(c>0)$ touches both the coordinate axes and lies in the third quadrant…

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