find the equation of circle which passes through the point(4,1)and(6,5)and whose center lies on the line 4x+y=16
Principal equation of circle states that
$\displaystyle (x-h)^2+(y-k)^2=r^2,$ where $\displaystyle (h,k)$ & $\displaystyle r$ are its center and radius respectively.
Since points $\displaystyle (4,1)$ & $\displaystyle (6,5)$ satisfy the equation, plug'em into the aforesaid equation.
Besides, its center lies on the line $\displaystyle 4x+y=16,$ so $\displaystyle 4h+k=16.$
You need to solve a system of equations.