Reciprocal Property of Pole and Polar
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Theorem
Let $\CC$ be a circle.
Let $P$ and $Q$ be points in the plane of $\CC$.
Let $\PP$ and $\QQ$ be the polars of $P$ and $Q$ with respect to $\CC$ respectively.
Let $Q$ lie on $\PP$ with respect to $\CC$.
Then $P$ lies on $\QQ$.
That is:
- if $P$ lies on the polar of $Q$, then $Q$ lies on the polar of $P$
- if the pole of $\PP$ lies on $\QQ$, then the pole of $\QQ$ lies on $\PP$
Proof
Let $\CC$ be a circle of radius $r$ whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_0, y_0}$.
Let $Q = \tuple {x_1, y_1}$.
The polar of $P$ is given by:
- $x x_0 + y y_0 = r^2$
The polar of $Q$ is given by:
- $x x_1 + y y_1 = r^2$
Let $Q$ lie on the polar of $P$.
Then $Q$ satisfies the equation:
- $x_0 x_1 + y_0 y_1 = r^2$
which is exactly the same as the condition for $P$ to lie on $Q$.
$\blacksquare$
Also see
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $8$. Reciprocal property of pole and polar