Question

A circle passes through the point $(1,2)$ and touches both the axes line Find its equation.

Answer 1

Let the equation of circle be $x^2+y^2+2gx+2fy+c=0$

Passes through $(1,2)⟹1+4+2g+4f+c=0\cdots \left(1\right)$

Touches $x-axis⟹g^2=c\cdots \left(2\right)$

Touches $y-axis⟹f^2=c\cdots \left(3\right)$

From $\left(2\right)\left(3\right)⟹g=f$

Put it in $\left(1\right)$

$5+2g+4g+g^2=0g^2+6g+5=0g^2+5g+g+5=0g(g+5)+1(g+5)=0(g+5)(g+1)=0g=-1,-5f=-1,-5c=1,25$

So the required equation of circles are $x^2+y^2-10x-10y+25=0x^2+y^2-2x-2y+1=0$