Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Conic sections

Example 6 from Algebra and Trigonometry, 12.5 Conic Sections in Polar Coordinates

Finding the Polar Form of a Horizontal Conic Given a Focus at the Origin and the Eccentricity and Directrix

Find the polar form of a conic given a focus at the origin, $e = \frac{3}{5},$ and directrix $x = 4.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Because the directrix is $x = p,$ we know the function in the denominator is cosine. Because $x = 4, 4 > 0,$ so we know there is an addition sign in the denominator. We use the standard form of

$r = \frac{e p}{1 + e \cos}$

and $e = \frac{3}{5}$ and $| 4 | = 4 = p .$

Therefore,

$\begin{array}{l} \begin{array}{l} r = \frac{(\frac{3}{5}) (4)}{1 + \frac{3}{5} \cos θ} \end{array} \\ r = \frac{\frac{12}{5}}{1 + \frac{3}{5} \cos θ} \\ r = \frac{\frac{12}{5}}{1 (\frac{5}{5}) + \frac{3}{5} \cos θ} \\ r = \frac{\frac{12}{5}}{\frac{5}{5} + \frac{3}{5} \cos θ} \\ r = \frac{12}{5} \cdot \frac{5}{5 + 3 \cos θ} \\ r = \frac{12}{5 + 3 \cos θ} \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the polar form of the conic given a focus at the origin, $e = 1,$ and directrix $x = −1.$

How did it go?

PreviousFinding the Polar Form of a Vertical Conic Given a Focus at the Origin and the Eccentricity and Directrix Next Converting a Conic in Polar Form to Rectangular Form

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 12.5, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.