Pair of Tangents from an External Point

Section 1.7 found the chord of contact joining the two points where tangents from an external point $P(x_1,y_1)$ touch a circle. It is also possible to write down a single combined equation for the two tangent lines themselves (as a pair, since together they form a second-degree curve — a pair of straight lines through $P$). The result is

where $S\equiv x^2+y^2+2gx+2fy+c$ is the circle's expression (now treated as a function of the running point $(x,y)$), $S_{11}$ is the power of $P$ (a fixed number, from §1.4), and $S_1\equiv xx_1+yy_1+g(x+x_1)+f(y+y_1)+c$ is the same mixed expression used for the tangent/polar/chord-of-contact formulas. Expanding this out always produces a second-degree equation that factorises (over the reals) into exactly the two tangent lines from $P$.

This combined equation is also the fastest route to the angle between the two tangents. If $\theta$ is that angle, it can be shown that $\tan\!\left(\dfrac{\theta}{2}\right)=\dfrac{r}{\sqrt{S_{11}}}$, where $r$ is the circle's radius — so the angle widens as $P$ gets closer to the circle (smaller $\sqrt{S_{11}}$) and narrows towards $0$ as $P$ moves far away.

Worked Example. Find the combined equation of the pair of tangents drawn from $(5,0)$ to the circle $x^2+y^2=9$. …