Replace the polar equation with an equivalent Cartesian equation.
-$$\mathrm { r } = 7 \cot \theta \csc \theta$$
A) $y = \frac { 7 } { x }$
B) $y ^ { 2 } = 7 x$
C) $y = 7 x$
D) $y = 7 x ^ { 2 }$

Question

Quizplus · Accepted Answer

We can use the identities $r \sin 	heta = y$ and $r \cos 	heta = x$ to convert the polar equation to a Cartesian equation as follows:
\begin{align*}
r &= 7 \cot 	heta \csc 	heta \
\sqrt{x^2 + y^2} &= 7 \frac{\cos 	heta}{\sin 	heta} \frac{1}{\sin 	heta} \
\frac{\sqrt{x^2 + y^2}}{x} &= \frac{7}{\sin^2 	heta} \
\left(\frac{\sqrt{x^2 + y^2}}{x}ight)^2 &= \left(\frac{7}{\sin^2 	heta}ight)^2 \
\frac{x^2 + y^2}{x^2} &= \frac{49}{\sin^4 	heta} \
\frac{x^2}{x^2 + y^2} &= \frac{\sin^4 	heta}{49} \
\frac{x^2}{7x^2 - 7y^2} &= 1 \
\frac{y^2}{x} &= \frac{1}{7} \
y^2 &= \frac{x}{7}
\end{align*}

Replace the Polar Equation with an Equivalent Cartesian Equation r=7cot⁡θcsc⁡θ\mathrm { r } = 7 \cot \theta \csc \thetar=7cotθcscθ

Replace the Polar Equation with an Equivalent Cartesian Equation $\mathrm { r } = 7 \cot \theta \csc \theta$