# Definition:Polar of Point/Ellipse

< Definition:Polar of Point(Redirected from Definition:Polar of Point wrt Ellipse)

Jump to navigation
Jump to search
This page has been identified as a candidate for refactoring of medium complexity.In particular: As a polar of a point wrt a conic section has been given a geometric definition, it makes sense to make this a Theorem page, as it should now be possible to demonstrate the truth of this analytically.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## 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: $3$.