To determine which polar equation represents an ellipse, we can refer to the standard forms of conic sections in polar coordinates. An ellipse can be represented in polar coordinates as:
\[ r = \frac{ed}{1 - e \cos \theta} \]
or
\[ r = \frac{ed}{1 - e \sin \theta} \]
where \(e\) (the eccentricity) is less than 1.
Analyzing the provided options:
-
\( r = \frac{3}{2 - 5 \sin \theta} \)
- Here, the denominator matches the form \(1 - e\sin \theta\), but \(5 > 2\), which means \(e = \frac{5}{2} > 1\), indicating that this is not an ellipse.
-
\( r = \frac{2}{2 - \sin \theta} \)
- This equation can be rewritten as \( r = \frac{2}{1 - \frac{1}{2} \sin \theta} \). So here, \( e = \frac{1}{2} < 1\), indicating that this is indeed an ellipse.
-
\( r = \frac{4}{3 + 3 \cos \theta} \)
- This form can be rearranged to \(r = \frac{4}{1 + \frac{3}{4} \cos \theta}\). Here, \( e = -\frac{3}{4} \), which does not define an ellipse since \(e\) should be positive.
-
\( r = \frac{5}{3 + 4 \cos \theta} \)
- Similarly, this can be represented as \(r = \frac{5}{1 + \frac{4}{5} \cos \theta}\); here \( e = -\frac{4}{5} \), which again does not represent an ellipse.
Thus, the polar equation that represents an ellipse is:
\( r = \frac{2}{2 - \sin \theta} \).