A circle x2+y2+4x-22 y + c = 0 is director circle of s1 and s1, is director of s2 and so on if the sum of radii of all these infinite number of circles is 2 then 2c =

Radius of the given circle is =4+2−C=6−Cr=2r1 and r1=2r2′r2=2r3Sum of radii of all given circles==r+r2+r2+…………=r1−12S∞=⇒r1−12=2⇒c=42

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A circle $x^{2} {+y}^{2} +4x-2 \sqrt{2} y + c = 0$ is director circle of s₁ and s₁, is director of s₂ and so on if the sum of radii of all these infinite number of circles is 2 then $\sqrt{2} c =$

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Detailed Solution

Radius of the given circle is $= \sqrt{4 + 2 - C} = \sqrt{6 - C}$

$r = \sqrt{2} r_{1} and r_{1} = \sqrt{2} r_{2^{'}} r_{2} = \sqrt{2} r_{3}$

Sum of radii of all given circles= $= r + \frac{r}{\sqrt{2}} + \frac{r}{2} + \dots \dots \dots \dots = \frac{r}{1 - \frac{1}{\sqrt{2}}}$

$S_{\infty} = \Rightarrow \frac{r}{1 - \frac{1}{\sqrt{2}}} = 2 \Rightarrow c = 4 \sqrt{2}$

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