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 conics polar coordinates

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > conics > polar form

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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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 solution 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