Let the circle be
 x^2+y^2+Dx+Ey+F=0

(2,1) :   2^2+1^2+E+F=0

(0,5) :   0^2+5^2+5E+F=0

(-1,2) :  (-1)^2+2^2-D+2E+F=0

Start with the General Form of circle.

 Substitution of P, Q, R. 

2D + E + F = -5 ….. (1)

5E + F = -25 ………(2)

D – 2E – F = 5 …….(3)

Simplification

(1) + (3),   3D – E = 0 ……. (4)

(2) + (3),   D + 3E = -20 ….. (5)

From (4),   E = 3D …………(6)

Subst. (6) in (5),  D + 9D = -20

    /\ D = -2 ………………. (7)

Subst. (7) in (6),  E = -6 …….. (8)

Subst. (7), (8) in (1),  F = -5

Solve the equations.

Eliminating F and get a simultaneous equation

in D and E.

Solve D and E

and finally get F.

The equation of the circle is

x^2+y^2 -2x -6y +5 = 0

Substitute D, E, F

in the General Form.