Definition:Polar of Point
Definition
Let $\KK$ be a conic section embedded in a Euclidean plane.
Let $P$ be an arbitrary point in that plane.
Let a secant line pass through $P$ and intersect $\KK$ at $L$ and $M$.
The polar of $P$ with respect to $\KK$ is the straight line upon which the tangents to $\KK$ intersect.
Circle
Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.
Let $P = \tuple {x_0, y_0}$ be an arbitrary point in the Cartesian plane.
The polar of $P$ with respect to $\CC$ is the straight line whose equation is given by:
- $x x_0 + y y_0 = r^2$
Ellipse
Let $\EE$ be an ellipse embedded in a Cartesian plane in reduced form with the equation:
- $\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$
Let $P = \tuple {x_0, y_0}$ be an arbitrary point in the Cartesian plane.
The polar of $P$ with respect to $\EE$ is the straight line whose equation is given by:
- $\dfrac {x x_0} {a^2} + \dfrac {y y_0} {b^2} = 1$
Also see
- Polar with respect to Conic Section forms Straight Line: demonstrating the straightness of the polar.
- Results about polars of points can be found here.