Question

Which polar equation represents an ellipse?

r= 1/3+2cos theta
r= 3/2+3 sin theta
r= 5/2+2 sin theta
r= 2/2-3 sin theta

Answer 1

Answer:

$r=\frac{1}{3+2cos\theta}$

Step-by-step explanation:

Let us write the equations in standard form:

$r=\frac{1}{3+2cos\theta} \implies r=\frac{\frac{1}{3} }{1+\frac{2}{3}\cos \theta }$

We have

$e=\frac{2}{3}\:<\:1$ and

 $ep=\frac{1}{3}$

Since the eccentricity  of this conic is less than 1, the conic represents an ellipse.

The second equation is $r=\frac{3}{2+3\sin \theta}$.

This is a hyperbola, because eccentricity is more than 1.

The third equation is $r=\frac{5}{2+2\sin \theta}$.

This is a parabola, because eccentricity is  1.

The fourth equation is $r=\frac{2}{2-3\sin \theta}$.

This is also a hyperbola, because eccentricity is more than 1.