Position of a Point Relative to a Circle (Power of a Point)

Given a circle $S\equiv x^2+y^2+2gx+2fy+c=0$ and any point $P(x_1,y_1)$ in the plane, a very useful quantity is obtained by simply substituting the point's coordinates into $S$:

This number is called the power of the point $P$ with respect to the circle, and its sign tells you exactly where $P$ sits:

• if $S_{11}<0$, $P$ lies inside the circle;
• if $S_{11}=0$, $P$ lies on the circle;
• if $S_{11}>0$, $P$ lies outside the circle.

The reasoning behind this is a straightforward distance comparison: $S_{11}$ turns out to equal $CP^2-r^2$, where $C$ is the centre and $r$ the radius. So $S_{11}<0$ exactly says $CP<r$ (closer to the centre than the radius, i.e. inside), and the other two cases follow the same way.

This same fact gives one of the most useful formulas in the whole chapter: if $P$ is outside the circle and $PT$ is a tangent segment from $P$ touching the circle at $T$, then by the Pythagorean theorem in the right triangle $CTP$ (the radius $CT$ is always perpendicular to the tangent at the point of contact), $PT^2=CP^2-CT^2=CP^2-r^2=S_{11}$. So the length of the tangent from an external point is simply

— no need to actually find the tangent line first; just plug the point into the circle's equation and take the square root.

Worked Example 1 (locating a point). Is the point $(3,-1)$ inside, on, or outside the circle $x^2+y^2-2x+4y-4=0$? …