$\text{ If }y=2x\text{ is a chord of the circle }$$x^2+y^2=10x,\text{ then the equation of }$$\text{ the circle whose diameter is this }$$\text{ chord, is - }$

$\begin{array}{l}\text{Equation of circle passing through the points of intersection of the given circle and the chord is }\\ x^2+y^2-10x+k\left(2x-y\right)=0\\ x^2+y^2+x\left(-10+2k\right)-ky=0\\ \text{center}\left(5-k,\frac{k}{2}\right)\\ \text{it lies on the line }y=2x\\ \frac{k}{2}=10-2k\\ k=20-4k\\ k=4\\ \text{Therefore, the equation of the circle is }\\ x^2+y^2+x\left(-10+8\right)-4y=0\\ x^2+y^2-2x-4y=0\end{array}$