Dear forum members,

I have a following problem

A circle touches both the x-axis and the y-axis and passes through the point (1,2). Find the equation of the circle. How many possibilities are there?

My solution

If the circle touches the x-axis and the y-axis, it means that both the x-axis and the y-axis are tangents to the circle, meaning that the distance from the centre to these tangents equals the radius, right? And the distance from the point (1,2) to the centre equals the radius as well.

Using a distance from a line formula I get that the distance from the center to the y-axis =(y)(Please pretend the parentheses are absolute value bars). The distance from the center to the x-axis is (x), pretend the parentheses are absolute value bars again.

that means (y)=(x)

if the center is marked as (x,y)

the distance from the point (1,2) to (x,y) is

$\displaystyle y^2=x^2-2x-4y+5=0 $ (I plugged in y instead of the radius)

My question is, is the above argument (y)=(x) good enough, for me to be able to substitute x to the place of y into the above equation?

Thank you in advance!