How do you convert r=4sec(theta)(1+ tan(theta)) into cartesian form?

Oct 26, 2016

$\frac{1}{4} = \frac{1}{x} + \frac{1}{y}$

Explanation:

Polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ are related as

$x = r \cos \theta$ and $y = r \sin \theta$, i.e. $\tan \theta = \frac{y}{x}$

Hence, $r = 4 \sec \theta \left(1 + \tan \theta\right)$ can be written as

$r \cos \theta = 4 \left(1 + \tan \theta\right)$

or $y = 4 \left(1 + \frac{y}{x}\right)$

or $x y = 4 x + 4 y$

or $\frac{x y}{4 x y} = \frac{4 x}{4 x y} + \frac{4 y}{4 x y}$

or $\frac{1}{4} = \frac{1}{y} + \frac{1}{x}$ or $\frac{1}{4} = \frac{1}{x} + \frac{1}{y}$