Chord of Contact

When a point $P(x_1,y_1)$ lies outside a circle, two distinct tangent lines can be drawn from $P$ to the circle, touching it at two points, say $Q$ and $R$. The line segment $QR$ joining those two points of contact is called the chord of contact of $P$ with respect to the circle.

Remarkably, the equation of this chord uses exactly the same algebraic recipe as the tangent equation from §1.5 — the 'replace $x^2\to xx_1$, $y^2\to yy_1$' substitution applied to $S=0$ using the coordinates of $P$:

The difference is purely in where $P$ sits: when $P$ is on the circle, $S_1=0$ is the tangent at $P$ itself; when $P$ is outside the circle, the very same formula $S_1=0$ instead gives the chord of contact — the line through the two (different) points where the tangents from $P$ touch. It's worth keeping this distinction straight, because the formula alone can't tell you which case you're in; you have to check the position of $P$ first (using $S_{11}$ from §1.4). …