## 17Calculus - Conics in Polar Coordinates

##### 17Calculus

As you learned on the main conics page, there is a standard equation for conics, i.e. $$Ax^2+Bxy+Cy^2+$$ $$Dx+Ey+F=0$$. Conics are particularly nice in polar coordinates and the equations are, in many ways, easier to represent and use.

The polar equation for a conic will be in one of these four forms.

$$\displaystyle{ r = \frac{ed}{1\pm e \sin\theta}}$$ e is the eccentricity $$\abs{d}$$ is the distance between the focus at the pole and its directrix

In order to write these equations in this form, we require that the focus (or one of the foci) be located at the origin. When we have a conic in this form, we can use the eccentricity to classify the equation, as follows.

eccentricity type ellipse parabola hyperbola

This video clip gives a nice overview of conic sections in polar coordinates and the presenter uses an example of a parabola to explain the equations.

### MIP4U - Graphing Conic Sections Using Polar Equations - Part 1 [4min-7secs]

video by MIP4U

To get a better understanding of these equations, we will look at examples of each of the three types of conics (parabolas, ellipses and hyperbolas). To help you understand these equations, get out a piece of paper and a pencil and do some calculations to convince yourself why these graphs look like they do.

Parabola

$$e=1$$

$$d=2$$

$$\displaystyle{r = \frac{2}{1+\sin(\theta)} }$$

$$e=1$$

$$d=2$$

$$\displaystyle{r = \frac{2}{1-\sin(\theta)} }$$

$$e=1$$

$$d=2$$

$$\displaystyle{r = \frac{2}{1+\cos(\theta)} }$$

$$e=1$$

$$d=2$$

$$\displaystyle{r = \frac{2}{1-\cos(\theta)} }$$

Practice

Find the polar equation of the parabola with vertex $$(4, 3\pi/2)$$.

Problem Statement

Find the polar equation of the parabola with vertex $$(4, 3\pi/2)$$.

Hint

You are not given all the information about this parabola in order to determine an exact parabola. So your final answer will have a variable of $$\theta$$ in it.

Problem Statement

Find the polar equation of the parabola with vertex $$(4, 3\pi/2)$$.

Hint

You are not given all the information about this parabola in order to determine an exact parabola. So your final answer will have a variable of $$\theta$$ in it.

Solution

### Krista King Math - 1610 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Ellipse

$$e=3/4$$

$$d=4/3$$

$$\displaystyle{r = \frac{1}{1+0.75\sin(\theta)} }$$

$$e=3/4$$

$$d=4/3$$

$$\displaystyle{r = \frac{1}{1-0.75\sin(\theta)} }$$

$$e=3/4$$

$$d=4/3$$

$$\displaystyle{r = \frac{1}{1+0.75\cos(\theta)} }$$

$$e=3/4$$

$$d=4/3$$

$$\displaystyle{r = \frac{1}{1-0.75\cos(\theta)} }$$

Here is a video of an application of an ellipse in polar coordinates. At this point, you should be able to understand the equations in this video. We hope you find it interesting to see an application of these equations.

### TU Delft Online Learning - The Trajectory Equation [14min-31secs]

Practice

Find the polar equation of the ellipse with eccentricity = $$1/2$$, directrix $$r=4 \sec \theta$$ and focus $$(0,0)$$.

Problem Statement

Find the polar equation of the ellipse with eccentricity = $$1/2$$, directrix $$r=4 \sec \theta$$ and focus $$(0,0)$$.

Solution

### Krista King Math - 2635 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

A conic section is given by the polar equation $$\displaystyle{ r = \frac{10}{3-2\cos\theta} }$$. Find the eccentricity, identify the conic, locate the directrix and sketch the conic.

Problem Statement

A conic section is given by the polar equation $$\displaystyle{ r = \frac{10}{3-2\cos\theta} }$$. Find the eccentricity, identify the conic, locate the directrix and sketch the conic.

Solution

### 2636 video solution

Log in to rate this practice problem and to see it's current rating.

Hyperbola

$$e=2$$

$$d=1$$

$$\displaystyle{r = \frac{2}{1+2\sin(\theta)} }$$

$$e=2$$

$$d=1$$

$$\displaystyle{r = \frac{2}{1-2\sin(\theta)} }$$

$$e=2$$

$$d=1$$

$$\displaystyle{r = \frac{2}{1+2\cos(\theta)} }$$

$$e=2$$

$$d=1$$

$$\displaystyle{r = \frac{2}{1-2\cos(\theta)} }$$

Practice

Find the polar equation of the hyperbola with eccentricity = $$1.5$$ and directrix $$y=2$$.

Problem Statement

Find the polar equation of the hyperbola with eccentricity = $$1.5$$ and directrix $$y=2$$.

Solution

### Krista King Math - 1611 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Determining the Type of Conic Section From the Equation

After studying the previous sets of graphs, you should have started to get a handle on how the graphs and equations are related. You will probably be asked to determine the type of conic from the equation. You already know that the eccentricity will help you a lot to determine the general type.

This video contains several examples, showing details on what to look for.

### MIP4U - Ex: Determine the Type of Conic Section Given a Polar Equation [4min-15secs]

video by MIP4U

Practice

Identify the conic, find the eccentricity and directrix and sketch the conic with equation $$\displaystyle{ \frac{9}{6+2\cos\theta} }$$.

Problem Statement

Identify the conic, find the eccentricity and directrix and sketch the conic with equation $$\displaystyle{ \frac{9}{6+2\cos\theta} }$$.

Solution

### Krista King Math - 1612 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Identify the conic given by the polar equation $$\displaystyle{ r = \frac{5}{10-15\sin \theta} }$$, then determine the directrix and eccentricity.

Problem Statement

Identify the conic given by the polar equation $$\displaystyle{ r = \frac{5}{10-15\sin \theta} }$$, then determine the directrix and eccentricity.

Solution

### PatrickJMT - 1613 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Write the polar equation of the conic with directrix $$x=3$$ and eccentricity = $$2/3$$.

Problem Statement

Write the polar equation of the conic with directrix $$x=3$$ and eccentricity = $$2/3$$.

Solution

### PatrickJMT - 1614 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.

Find the polar equation of the ellipse with eccentricity = $$1/2$$ and directrix $$r = \sec \theta$$.

Problem Statement

Find the polar equation of the ellipse with eccentricity = $$1/2$$ and directrix $$r = \sec \theta$$.

Solution

### Krista King Math - 1615 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

Graph $$\displaystyle{ r = \frac{8}{2-2\cos\theta} }$$ and label all key components.

Problem Statement

Graph $$\displaystyle{ r = \frac{8}{2-2\cos\theta} }$$ and label all key components.

Solution

### MIP4U - 1616 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Graph $$\displaystyle{ r = \frac{8}{4+2\sin\theta} }$$ and label all key components.

Problem Statement

Graph $$\displaystyle{ r = \frac{8}{4+2\sin\theta} }$$ and label all key components.

Solution

### MIP4U - 1617 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Graph $$\displaystyle{ r = \frac{8}{2-4\sin\theta} }$$ and label all key components.

Problem Statement

Graph $$\displaystyle{ r = \frac{8}{2-4\sin\theta} }$$ and label all key components.

Solution

### MIP4U - 1618 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find the intercepts and foci of $$\displaystyle{ r = \frac{4}{4-2\cos\theta} }$$.

Problem Statement

Find the intercepts and foci of $$\displaystyle{ r = \frac{4}{4-2\cos\theta} }$$.

Solution

### MIP4U - 1619 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find the intercepts and foci of $$\displaystyle{ r = \frac{6}{3-3\sin\theta} }$$.

Problem Statement

Find the intercepts and foci of $$\displaystyle{ r = \frac{6}{3-3\sin\theta} }$$.

Solution

### MIP4U - 1620 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

Find the intercepts and the foci of $$\displaystyle{ r = \frac{12}{2+6\cos\theta} }$$.

Problem Statement

Find the intercepts and the foci of $$\displaystyle{ r = \frac{12}{2+6\cos\theta} }$$.

Solution

### MIP4U - 1621 video solution

video by MIP4U

Log in to rate this practice problem and to see it's current rating.

### polar conics 17calculus youtube playlist

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.