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

`r=(1)/(3+2cos\theta)`

Explanation:

Let us write the equations in standard form:

`r=(1)/(3+2cos\theta) \implies r=((1)/(3) )/(1+(2)/(3)\cos \theta )`

We have

`e=(2)/(3)\:<\:1` and

`ep=(1)/(3)`

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

The second equation is
`r=(3)/(2+3\sin \theta)`.

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

The third equation is
`r=(5)/(2+2\sin \theta)`.

This is a parabola, because eccentricity is 1.

The fourth equation is
`r=(2)/(2-3\sin \theta)`.

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