## 17Calculus - Conics in Polar Coordinates

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 $$e=1$$

$$d=2$$

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

$$d=2$$

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

$$d=2$$

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

$$d=2$$

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

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

Problem Statement

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

Solution

### 1610 video

video by Krista King Math

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]

Hyperbola $$e=2$$

$$d=1$$

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

$$d=1$$

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

$$d=1$$

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

$$d=1$$

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

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

### 1611 video

video by Krista King Math

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

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

### 1612 video

video by Krista King Math

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

### 1613 video

video by PatrickJMT

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

### 1614 video

video by PatrickJMT

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

### 1615 video

video by Krista King Math

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

### 1616 video

video by MIP4U

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

### 1617 video

video by MIP4U

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

### 1618 video

video by MIP4U

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

### 1619 video

video by MIP4U

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

### 1620 video

video by MIP4U

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

video by MIP4U

### polar conics 17calculus youtube playlist

You CAN Ace Calculus

 conics polar coordinates

### Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem. The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

You Can Have an Amazing Memory: Learn Life-Changing Techniques and Tips from the Memory Maestro Under Armour Clothing - Just Launched at eBags.com! Prime Student 6-month Trial 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.