Q.Find the equation(s) of the circle(s) which touch the $x$-axis at the point $(3,0)$ and have radius $5$.

A circle is the locus of points at a fixed distance (the radius $r$) from a fixed centre $(h,k)$; applying the distance formula to this definition gives the standard equation $(x-h)^2+(y-k)^2=r^2$, which reduces to $x^2+y^2=r^2$ when the centre is the origin. Expanding the standard form produces the general equation $x^2+y^2+2gx+2fy+c=0$, from which the centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$ can always be recovered, either by direct substitution into these formulas or by completing the square. The sign of $g^2+f^2-c$ determines whether the equation represents a genuine circle (positive), degenerates to a sin …