The radius of a circle touching all the four | vedclass

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(B) The four circles are centered at $(\lambda, \lambda), (\lambda, -\lambda), (-\lambda, \lambda),$ and $(-\lambda, -\lambda)$ with radius $\lambda$.

Vedclass · Accepted Answer

$(\sqrt{2} - 1) \lambda$

Vedclass · Answer

(B) The four circles are centered at $(\lambda, \lambda), (\lambda, -\lambda), (-\lambda, \lambda),$ and $(-\lambda, -\lambda)$ with radius $\lambda$.Let the required circle be centered at the origin $(0, 0)$ with radius $r$.The distance from the origin to the center of any of the four circles is $\sqrt{\lambda^2 + \lambda^2} = \sqrt{2} \lambda$.Since the required circle touches these four circles externally,the distance between the center of the required circle and the center of any of the four circles must be equal to the sum of their radii,i.e.,$r + \lambda$.Therefore,$r + \lambda = \sqrt{2} \lambda$.Solving for $r$,we get $r = \sqrt{2} \lambda - \lambda = (\sqrt{2} - 1) \lambda$.

The radius of a circle touching all the four circles $(x \pm \lambda)^2 + (y \pm \lambda)^2 = \lambda^2$ is

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