Polar of Point is Perpendicular to Line through Center
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Theorem
Let $\CC$ be a circle.
Let $P$ be a point.
Let $\LL$ be the polar of $P$ with respect to $\CC$.
Then $\LL$ is perpendicular to the straight line through $P$ and the center of $\CC$.
Proof
Let $\CC$ be positioned so as for its center to be at the origin of a Cartesian plane.
Let $P$ be located at $\tuple {x_0, y_0}$.
From Equation of Straight Line in Plane, $P$ can be described as:
- $y = \dfrac {y_0} {x_0} x$
and so has slope $\dfrac {y_0} {x_0}$.
By definition of polar, $\LL$ has the equation:
- $x x_0 + y y_0 = r^2$
which has slope $-\dfrac {x_0} {y_0}$.
Hence the result from Condition for Straight Lines in Plane to be Perpendicular.
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $6$.