Definition:Polar of Point/Ellipse
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Definition
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:
- $\dfrac {x x_0} {a^2} + \dfrac {y y_0} {b^2} = 1$
Pole
Let $\LL$ be the polar of $P$ with respect to $\EE$.
Then $P$ is known as the pole of $\LL$.
Also see
- Definition:Chord of Contact on Ellipse: when $P$ is specifically outside $\EE$
- Results about polars of points can be found here.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $3$.