The standard form of the equation of a circle

(x – a)^2 + (y – b)^2 = r^2, where the point (a,b) is the center and r is the radius of the circle

(4,1) lies on the circle so (4 - a)^2 + (1 - b)^2 = r^2

(6,5) lies on the circle so (6 - a)^2 + (5 - b)^2 = r^2

(a, b) the center of the circle lies on the line 4x + y = 16 and thus 4a + b = 16

we have system of 3 equations

(4 - a)^2 + (1 - b)^2 = r^2

(6 - a)^2 + (5 - b)^2 = r^2

4a + b = 16

by solving the system we find

a = 3

b = 4

r = 10^(1/2)

the equation of the circle is

(x - 3)^2 + (y - 4)^2 = 10