Q.Find the equation of the circle passing through the points $(0,0)$, $(4,0)$, and $(0,3)$.

With three points given, substitute each into the general equation of a circle to get three simultaneous equations in the unknowns $g$, $f$, $c$, then solve them together.

Let the circle be $x^2+y^2+2gx+2fy+c=0$. Substitute each given point in turn.

Point $(0,0)$: $0+0+0+0+c=0 \Rightarrow c=0$.

Point $(4,0)$: $16+0+8g+0+c=0$. With $c=0$: $16+8g=0 \Rightarrow g=-2$.

Point $(0,3)$: $0+9+0+6f+c=0$. With $c=0$: $9+6f=0 \Rightarrow f=-\dfrac32$.

So the circle is:

$x^2+y^2+2(-2)x+2\left(-\dfrac32\right)y+0=0 \quad \Rightarrow \quad x^2+y^2-4x-3y=0$ …

Forgetting that $c=0$ once found must be carried into the next two substitutions is a common oversight, leading to an extra unknown term that never gets solved. Always substitute the already-found values immediately, and finish by chec …