Ellipse in polar coordinates I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - 

Where $2a$ is the total height of the ellipse and $2b$ being the total width.
You can then find the radial length, $r$, at any angle $\theta$ to major axis as...
$$r(\theta) = \sqrt{(b \sin(\theta))^2 + (a \cos(\theta))^2}$$
...by just following the Pythagorean theorem. Yet Wikipedia's equation for the polar coordinate ellipse is as follows:
$$r(\theta) = \frac{ab}{\sqrt{(b \cos(\theta))^2 + (a \sin(\theta))^2}}$$
Here is the link to the Wikipedia page: 
Can someone explain this, please? Why divide by the hypotenuse? Why the $ab$? Thank you!

Polar Equation from the Center of the Ellipse
The equation of an ellipse is
$$
\left(\frac{x}{a}\right)^2+\left(\frac{y}{a\sqrt{1-e^2}}\right)^2=1\tag1
$$
Using $x=r\cos(\theta)$ and $y=r\sin(\theta)$ in $(1)$, we get
$$
r^2\cos^2(\theta)+\frac{r^2\sin^2(\theta)}{1-e^2}=a^2\tag2
$$
and we can solve $(2)$ for $r^2$ to get the polar equation
$$
r^2=\frac{\overbrace{a^2\!\left(1-e^2\right)}^{b^2}}{1-e^2\cos^2(\theta)}\tag3
$$


Polar Equation from a Focus of the Ellipse
Centered at the right focus
$$
\left(\frac{x+ae}a\right)^2+\left(\frac{y}{a\sqrt{1-e^2}}\right)^2=1\tag4
$$
Using $x=r\cos(\theta)$ and $y=r\sin(\theta)$ in $(4)$, we get
$$
r^2\cos^2(\theta)+2aer\cos(\theta)+a^2e^2+\frac{r^2\sin^2(\theta)}{1-e^2}=a^2\tag5
$$
which gives the quadratic equation in $r$:
$$
\frac{r^2\left(1-e^2\cos^2(\theta)\right)}{1-e^2}+2aer\cos(\theta)-a^2\!\left(1-e^2\right)=0\tag6
$$
whose solution is
$$
r=\frac{a\!\left(1-e^2\right)}{1+e\cos(\theta)}\tag7
$$

