A line segment A B of length λ moves such that the points A and B remain on the periphery of a circle…

Given,

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$,

Now taking the points on the circle of radius $\lambda$ as $B\left(\lambda \cos {\theta }_1,\lambda \sin {\theta }_1\right) \& A\left(\lambda \cos {\theta }_2\lambda \sin {\theta }_2\right)$ and taking the point $P\left(h,k\right)$ which divides the line segment in $AB$ of length $\lambda$ in $2:3$

Now plotting the diagram we get,

Now, let $O$ be the origin and radius of circle is $\lambda$ and  $AB=\lambda$ and using distance formula we get,

$AB=\lambda =\sqrt{{\left(\lambda \cos {\theta }_1-\lambda \cos {\theta }_2\right)}^2+{\left(\lambda \sin {\theta }_1-\lambda \sin {\theta }_2\right)}^2}$

$\Rightarrow 1=2-2 \cos \left({\theta }_1-{\theta }_2\right)$

$\Rightarrow \cos \left({\theta }_1-{\theta }_2\right)=\frac{1}{2}$

Now using section formula we get,

$h=\frac{2\lambda \cos {\theta }_1+3\lambda \cos {\theta }_2}{5}$ and  $k=\frac{2\lambda \sin {\theta }_1+3\lambda \sin {\theta }_2}{5}$

Now squaring and adding above two value we get,

$h^2+k^2=\frac{{\lambda }^2}{25}\left[4+9+12\left(\cos \left({\theta }_1-{\theta }_2\right)\right)\right]$

$\Rightarrow h^2+k^2=\frac{{\lambda }^2}{25}·19$

Hence, Radius $=\frac{\lambda }{5}\sqrt{19}$