Given: Circle passes through (3, 4) and touch y-axis at origin, this means it also passes though origin O(0, 0).

To Find: Equation of Circle

General equation of Circle: x² + y² + 2gx + 2fy + c = 0

As circle passes through (0, 0). This point will satisfy the equation.

Therefore,

(0)² + (0)² + 2g(0) + 2f(0) + c = 0

c = 0

Now, we have the equation of circle as,

x² + y² + 2gx + 2fy = 0

Now, since the centre is on the x-axis, y coordinate of centre = 0. i.e. f = 0.

Therefore,

x² + y² + 2gx = 0

It is also given that this circle passes through (3, 4). So,

(3)² + (4)² + 2g(3) = 0

9 + 16 + 6g = 0

25 + 6g = 0

6g = -25

Hence, we have the equation as,