The Normal to a Circle

At any point $P$ on a circle, the normal is the line through $P$ that is perpendicular to the tangent at $P$. Because a tangent is always perpendicular to the radius drawn to the point of contact, the normal at $P$ is simply the line through $P$ and the centre of the circle — every normal to a circle passes through its centre. This makes writing its equation almost immediate: it's just the line through the two points $P(x_1,y_1)$ and the centre $(-g,-f)$.

Working out the two-point form of that line and simplifying gives the standard formula for the equation of the normal at $P(x_1,y_1)$ on the circle $S\equiv x^2+y^2+2gx+2fy+c=0$:

For the special case of a circle centred at the origin, $x^2+y^2=r^2$, this reduces to the very simple $xy_1-yx_1=0$ — just the line joining the origin to $(x_1,y_1)$, as you would expect.

Worked Example. Find the normal at the point $(6,2)$ on the circle $x^2+y^2-2x-4y-20=0$. …