To find an equation equivalent to the polar equation [tex]\( r = 7 \csc \theta \)[/tex] in rectangular coordinates and describe the graph of the equation, let's follow these steps:
1. Recall the polar to rectangular conversion formulas:
- [tex]\( r = \sqrt{x^2 + y^2} \)[/tex]
- [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex]
- [tex]\(\sin \theta = \frac{y}{r}\)[/tex]
2. Rewrite the given polar equation [tex]\( r = 7 \csc \theta \)[/tex]:
[tex]\[
r = 7 \cdot \frac{1}{\sin \theta}
\][/tex]
Substituting [tex]\(\sin \theta = \frac{y}{r}\)[/tex], we have:
[tex]\[
r = \frac{7}{\frac{y}{r}} \implies r = \frac{7r}{y}
\][/tex]
3. Simplify the equation:
[tex]\[
r^2 = 7r / y \implies r^2 y = 7r
\][/tex]
Since [tex]\(r \neq 0\)[/tex], we can divide both sides by [tex]\(r\)[/tex]:
[tex]\[
r y = 7
\][/tex]
4. Convert [tex]\(r\)[/tex] to rectangular coordinates:
[tex]\[
\sqrt{x^2 + y^2} \cdot y = 7
\][/tex]
Hence, the rectangular form of the equation is:
[tex]\[
y \cdot \sqrt{x^2 + y^2} = 7
\][/tex]
This equation represents a circle centered at [tex]\((0, \frac{7}{2})\)[/tex] with a radius of [tex]\(\frac{7}{2}\)[/tex]. Therefore, the graph of the equation is correctly described by option C.
Correct answer: C
