Pole and Polar of an Ellipse
In geometry, the concepts of pole and polar are powerful tools used to analyze curves in polar coordinates. The pole of a curve is a point with specific geometric properties related to that curve, and the polar is a curve associated with a given point, often the pole. Let's delve into how these concepts apply to an ellipse.
Pole of an Ellipse:
The pole of an ellipse in polar coordinates is a point that corresponds to the center of the ellipse in Cartesian coordinates. Denoted as $P$, the pole has coordinates $(h,k)$, where $(h,k)$ is the center of the ellipse in Cartesian coordinates.
The pole is a fundamental point because the polar coordinates of any point on the ellipse can be expressed in terms of its distance from the pole ($r$) and the angle formed with the positive xaxis ($\theta $). The relationship is given by:
$r(\theta )=\frac{a\cdot b}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}(\theta )+{b}^{2}{\mathrm{cos}}^{2}(\theta )}}$
where $a$ and $b$ are the semimajor and semiminor axes, respectively.
Polar of an Ellipse:
The polar of a point with respect to the ellipse is a curve that represents all possible points for which the given point is the pole. For the ellipse, the polar equation is derived from the relationship mentioned above:
$r(\theta )=\frac{a\cdot b}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}(\theta )+{b}^{2}{\mathrm{cos}}^{2}(\theta )}}$This equation defines the locus of points in polar coordinates for a specific pole, in this case, the center of the ellipse.
Special Cases:

Circle: If the ellipse is a circle ($a=b$, the polar equation simplifies to $r(\theta )=\frac{a}{\sqrt{2}}$. The pole would be at the center of the circle.

Horizontal Ellipse: When the major axis is along the xaxis ($a>b$), the polar equation becomes $r(\theta )=\frac{a\cdot b}{\sqrt{{a}^{2}{\mathrm{sin}}^{2}(\theta )+{b}^{2}{\mathrm{cos}}^{2}(\theta )}}$

Vertical Ellipse: When the major axis is along the yaxis ($b>a$), the polar equation is $r(\theta )=\frac{a\cdot b}{\sqrt{{a}^{2}{\mathrm{cos}}^{2}(\theta )+{b}^{2}{\mathrm{sin}}^{2}(\theta )}}$