The radius of a circle, having minimum area, which touches the...

Length of perpendicular from centre of circle to tangent line $r = \dfrac{k}{\sqrt{2}}$

And $r = \left|4 - k\right|$



$⇒ r = 4 - k = \dfrac{k}{\sqrt{2}}$

$⇒ 4 = \dfrac{k + \sqrt{2}k}{\sqrt{2}}$

$⇒ k = \dfrac{4\sqrt{2}}{\sqrt{2} + 1}$

$⇒ r = \dfrac{4\sqrt{2}}{\left(\sqrt{2} + 1\right)\sqrt{2}}$

$⇒ r = \dfrac{4}{\sqrt{2} + 1}$

$⇒ r = 4\left(\sqrt{2} - 1\right)$