Equation of Circle

Let the equation of the circle(s) be x2+ y2 + Dx + Ey + F = 0

Centre of the circle = (-D/2 , -E/2)

Radius of the circle = 1/2 √(D2 + E2 - 4F)

As the circle passes through (0 , 0),

0 + 0 + 0 + 0 + F = 0

we get F = 0

As the circle passes through (3 , 0)

(3)2 + 0 + 3D + 0 = 0

we get, D = -3

So, the radius of the circle = 1/2 √(9 + E2)

Centre of the circle = (3/2 , -E/2)

Now, as the circle touches the line x + y + 1 = 0

So,

│[3/2 - E/2 + 1]/√(12 + 12)│ = 1/2√(9 + E2)

(5/2 - E/2)2 / 2 = 1/4 (9 + E2)

25/8 - 5E/4 + E2/8 = 9/4 + E2/4

E2/8 + 5E/4 - 7/8 = 0

E2 + 10 - 7 = 0

E = {-10 +- 8√2} / 2 = (-5 +- 4√2)

So, the possible equation of circles are:

x2 + y2 - 3x + (-5 + 4√2)y = 0

or x2 + y2 - 3x - (5 + 4√2)y = 0