Identify the Conic With a Given Polar Equation - Pre-Calculus

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

Now, for the given conic section, so it must be an ellipse.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be an ellipse.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a parabola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a hyperbola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

Now, for the given conic section, so it must be a parabola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a hyperbola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a hyperbola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be an ellipse.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a parabola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be a parabola.

Recall that the polar equations of conic sections can come in the following forms:

, where is the eccentricity of the conic section.

To determine what conic section the polar graph depicts, look only at the conic section's eccentricity.

will give an ellipse.

will give a parabola.

will give a hyperbola.

First, put the given polar equation into one of the forms seen above by dividing everything by .

Now, for the given conic section, so it must be an ellipse.

This would be an ellipse.

The polar form of any conic is \[or cosine\], where e is the eccentricity. If the eccentricity is between 0 and 1, then the conic is an ellipse, if it is 1 then it is a parabola, and if it is greater than 1 then it is a hyperbola. Circles have eccentricity 0.

To figure out what the eccentricity is, we need to get our equation so that the denominator is in the form . Right now it is , so multiply top and bottom by :

.

Now we can identify our eccentricity as which is between 0 and 1.

All polar forms of conic equations are in the form \[or cosine\] where e is the eccentricity.

If the eccentricity is between 0 and 1 the conic is an ellipse, if it is 1 then it is a parabola, if it is greater than 1 it is a hyperbola. Circles have an eccentricity of 0.

We want the denominator to be in the form of , so we can multiply top and bottom by one half:

The eccentricity is 1, so this is a parabola.

Although it's not immediately obvious, this is a circle. One way we can see this is by converting from polar form to cartesian:

multiply both sides by r

we can now replace with and with :

We can already mostly tell this is a circle, but just to be safe we can put it all the way into standard form:

complete the square by adding to both sides

condense the left side

Now this is clearly a circle.