Convert the polar equation to rectangular form.

Question

Convert the polar equation to rectangular form. ​  ​

Quizplus · Accepted Answer

To convert a polar equation to rectangular form we use the following substitution:
$r = \sqrt{x^2+y^2}$ and $	heta = \arctan(\frac{y}{x})$.
Substituting $r=11e^{ac}\cos{	heta}$ and $r=11e^{ac}\sin{	heta}$ we get:
$x=11e^{ac}\cos{	heta}\cos{	heta} = 11e^{ac}\cos^2{	heta}$ and 
$y=11e^{ac}\sin{	heta}\cos{	heta} = \frac{11e^{ac}\sin{2	heta}}{2}$.
Simplifying the expression for $y$, we get:
$2y = 11e^{ac}\sin{2	heta} = 11e^{ac}2\sin{	heta}\cos{	heta} = 11e^{ac}\frac{y}{x} \frac{x}{\sqrt{x^2+y^2}}$.
Solving for $\frac{y}{x}$, we get:
$\frac{y}{x} = \frac{2}{\frac{11e^{ac}}{\sqrt{x^2+y^2}}} = \frac{2\sqrt{x^2+y^2}}{11e^{ac}}$.
Substituting this expression for $\frac{y}{x}$ in the expression for $x$, we get:
$x = 11e^{ac}\cos^2{	heta}$ and $y=\frac{2}{11}(\sqrt{x^2+y^2})e^{ac}\sqrt{1-\cos^2{	heta}}$.
Squaring and simplifying we obtain the rectangular equation:
$x^2 +(y-\frac{11}{2}e^{ac})^2 = \frac{121}{4}e^{2ac}$. 
Therefore, the correct choice is C).

Convert the Polar Equation to Rectangular Form