A circle $C$ touches the line $x=2y$ at the point $\left(2,1\right)$ and intersects the circle ${\mathrm{C}}_1:x^2+y^2+2y-5=0$ at two points $P$ and $Q$ such that $PQ$ is a diameter of $C_1$. Then the diameter of $C$ is :

Family of circle touching line $2y=x$ at point $\left(2,1\right)$
$(x-2{)}^2+(y-1{)}^2+\lambda (x-2y)=0 ...\left(1\right)$
common chord $PQ$ is
$(x-2{)}^2+(y-1{)}^2+\lambda (x-2y)-x^2-y^2-2y+5=0$
(diameter of $C_1$ passes through $(0,-1)$)
$4+4+2\mathrm{\lambda }-1+2+5=0$

$\mathrm{\lambda }=-7$ put in equation $\left(1\right)$

$(x-2{)}^2+(y-1{)}^2-7(x-2y)=0$

$x^2+y^2-11x+12y+5=0$

$r=\sqrt{\frac{121}{4}+36-5}=\sqrt{\frac{121+124}{4}}=\sqrt{\frac{245}{4}}=\frac{\sqrt{245}}{2}\Rightarrow \text{ diameter }=\sqrt{245}$$=7\sqrt{5}$