Pole and Polar, Conjugate Points and Lines, Inverse Points

The chord-of-contact idea generalises into one of the more elegant constructions in the chapter. Let $S=0$ be a circle and $P$ any point in the plane other than the centre. Draw any line (a secant) through $P$ that meets the circle in two points, and at each of those two points draw the tangent to the circle; those two tangents meet at some point. As the secant through $P$ is rotated, the intersection point of the two tangents traces out a straight line — that traced-out line is called the polar of $P$, and $P$ itself is called the pole of that line. Just as with the chord of contact, the equation of the polar of $P(x_1,y_1)$ with respect to $S=0$ is $S_1=0$ — the identical formula. When $P$ is outside the circle, its polar is exactly its chord of contact; when $P$ is inside, the polar still exists as a line (even though no real tangents can be drawn from $P$ to touch the circle).

A pleasant symmetry, called reciprocity, follows: the polar of $P$ passes through a point $Q$ if and only if the polar of $Q$ passes through $P$. When this mutual relationship holds, $P$ and $Q$ are called conjugate points with respect to the circle, and the condition for it is simply

(the same 'mixed' substitution, now using both points at once). Extending the idea one step further, if $P$ and $Q$ are conjugate points, their two polar lines are called conjugate lines; for two lines $l_1x+m_1y+n_1=0$ and $l_2x+m_2y+n_2=0$ to be conjugate with respect to a circle of radius $r$, the condition is $r^2(l_1l_2+m_1m_2)=(l_1g+m_1f-n_1)(l_2g+m_2f-n_2)$.

A separate but related notion is that of inverse points. Two points $P$ and $Q$ are inverse points with respect to a circle of centre $C$ and radius $r$ if $C$, $P$, $Q$ are collinear, $P$ and $Q$ lie on the same side of $C$, and $CP\cdot CQ=r^2$. Geometrically, inverse points are exactly what you get by intersecting the line $CP$ with the polar of $P$ — the polar of $P$ crosses the line joining $P$ to the centre precisely at $P$'s inverse point.

Worked Example 1 (polar). Find the polar of $(2,3)$ with respect to $x^2+y^2=4$.

Here $g=f=0,c=-4$, so the polar is $S_1=2x+3y-4=0$. …