To determine the polar equation, first we need to interpret the original cartesian graph. This is an ellipse with a vertical major axis with half its length . The minor axis has half its length . To find the foci, use the relationship

so

Since the center is , this means that the foci are at

Having a focus at the origin means we can use the formula  [or sine] where e is the eccentricity, and for an ellipse .

In this case, the focus at the origin is above the directrix, so we be subtracting. The major axis is vertical, so we are using sine.

Solving for p gives us:

The eccentricity is , in this case

This gives us an equation of:

We can simplify by multiplying top and bottom by 2:

To convert this cartesian equation to polar form, we will use the substitutions and .

First, we should expand the expression:

square x - 3

subtract 9 from both sides

group the squared variables next to each other - this will help see how to re-write this in polar form:

now make the substitutions:

This is a quadratic in r with

Solve using the quadratic formula:

Subtracting which would give us a circle with no radius, but adding so our answer is:

The directrix is to the left of the focus at the origin, and the major axis is horizontal, so our equation is going to take the form

where e is the eccentricity. For a hyperbola specifically,

First we need to solve for c so we can find the eccentricity. For a hyperbola, we use the relationship where is half the length of the minor axis and is half the length of the major axis, in this case the horizontal one.

In this case, and

add 9 to both sides

take the square root

To find the eccentricity: so in this case

gives us

Plugging in e and p:

To simplify we can multiply top and bottom by 3:

This is the equation for a parabola, so the eccentricity is 1. It opens up, so the focus is above the directrix.

This means that our equation will be in the form

where a is the distance from the focus to the vertex.

In this case, we have , so .

Because the vertex is , the focus is so we can use the formula without adjusting anything.

The equation is

.

This is the equation for a parabola with a vertex at . This parabola opens left because it is negative. The distance from the focus to the vertex can be found by solving , so . This places the focus at the origin.

Because of this and the fact that the focus is to the left of the directrix, we know that the polar equation is in the form

.

That means that this equation is

.

To put this in polar form, we need to understand its structure. We can find the foci by using the relationship where a is half the length of the major axis, and b is half the length of the minor axis.

In this problem,

.

Plugging these values into the above realationship we can solve for .

 divide by -1

Since the center is , the foci are at and .

The major axis is vertical, and the lower focus is at the origin, so the polar equation will be in the form where and for an ellipse, .

Here,  and 

Simplify the top by cancelling out 

 Simplify by multiplying top and bottom by 

Before writing this in polar form, we need to know the structure of this hyperbola. We can determine the foci by using the relationship

In this case, subtract 27

The center is at and the major axis is horizontal, so the foci are and , adding or subtracting 6 from the x-coordinate. Because the left focus is at the origin and the major axis is horizontal, our polar equation will be in the form where and for a hyperbola

Here,

Plugging this in gives:

simplify the numerator by cancelling the 2's

simplify once more by multiplying top and bottom by

This is the equation for a right-opening parabola with a vertex at . The distance from the vertex to the focus can be found by solving , so . This places the focus at .

Because the focus is at the origin, and the parabola opens to the right, this equation is in the form .

This particular parabola has a polar equation .

This is the equation for a down-opening parabola. The vertex is at . We can figure out the location of the focus by solving . This means that , so the focus is at .

Because the focus is at the origin, and the parabola opens down, the polar form of the equation is .

This equation is

.

First, multiply out :

we can re-arrange this a little bit by subtracting 2 from both sides and putting next to :

Now we can make the substitutions and :

We can solve for r using the quadratic formula:

factor out 4 inside the square root