How to find the equation of the circle touching the $y$-axis given that it passes through two particular points?
Hint: The center of the circle will be equidistant from the $y$ axis and either of the two centers. The points equidistant from the $y$ axis and one of the points will be a parabola; its equation is easy to find. Now you have to find the intersection points between two parabolas, which will involve solving a quadratic equation.
The Two points must be on the same side with respect to y axis. So I will take other example and that is when those two points are (1,3) and (2,4). Find the equations of all possible circles.
Step 1) Keep in mind that when a circle touch y axis, its radius is of the same value as its center's x-coordinate. (r=absolute (X center) ).
Step 2) Therefore Equation will be of the form (X-a)^2+(Y-b)^2=a^2 and we have to find a and b
Step 3) Substitute the two points int the equation:
(1-a)^2+(3-b)^2=a^2 and (2-a)^2+(4-b)^2=a^2 Solve the system. we get (a,b)=(1,4) or (5,0)