The equation of the circle passing through ( 4 , $1)$ and $(6,5)$ and whose centre is on the line $4

Question

The equation of the circle passing through ( 4 , $1)$ and $(6,5)$ and whose centre is on the line $4 x+y=16$ is

Professor PC Thomas Classes · Answer

Let the centre be $(h,k)$. Since it passes through $(4,1)$ and $(6,5)$, $(h-4)^2+(k-1)^2 = (h-6)^2+(k-5)^2 \implies 4h+8k = 44 \implies h+2k=11$. Also $4h+k=16$. Solving yields $h=3, k=4$. Radius squared $= (4-3)^2+(1-4)^2 = 10$. Eq: $(x-3)^2+(y-4)^2=10 \implies x^2+y^2-6x-8y+15=0$.