Find the equation of a circle passing through the point (1,2) and (3,4) and touching line 3x+y-3=0?

The equation of a circle is x^2 + y^2 = r^2. That is centered on the origin. If you want it centered on a point (a, b) then you subtract the coordinates:

(x - a)^2 + (y - b)^2 = r^2

You are given three solutions but you only need two of them to start:

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

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

r is a constant, so set the two equations equal:

(1 - a)^2 + (2 - b)^2 = (3 - a)^2 + (4 - b)^2

This is a very time consuming exercise and at least two answers. If the circle is allowed to touch at two points, there can be any number of answers. So the question needs to be clarified. One thing you would want to clarify is "Why do we need to work on such a complex situation?"