Find the equation of the circle through the points (0, 0), ( 2, 0) and ( 0, 4). Find the co-ordinates of its center and its radius?

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

It is given that the circle passes through the points  (0, 0), ( 2, 0) and ( 0, 4)

$$\begin{gathered} \mathrm{Substituting} \left(0,0\right) \mathrm{in} \mathrm{the} \mathrm{above} \mathrm{equation}, \mathrm{we} \mathrm{get} \\ \mathrm{c}=0 \\ \mathrm{Substituting} \left(2,0\right) \mathrm{in} \mathrm{the} \mathrm{above} \mathrm{equation}, \mathrm{we} \mathrm{get} \\ 4+0+4\mathrm{g}+\mathrm{c}=0, \\ 4 + 4\mathrm{g} + 0 = 0 \\ \mathrm{g} = -1 \\ \mathrm{Substituting} \left(0,4\right) \mathrm{in} \mathrm{the} \mathrm{above} \mathrm{equation}, \mathrm{we} \mathrm{get} \\ 0+16+0+8\mathrm{f}+\mathrm{c}=0 \\ \mathrm{f} = -2 \end{gathered}$$

So, the equation of the circle is $x^2+y^2-2x-4y=0$

Centre = $\left(-g,-f\right)=\left(1,2\right)$

Radius = $\sqrt{g^2+f^2-c}=\sqrt{1+4}=\sqrt{5}$