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Conics in polar form practice problems

A conic r = ed/(1 ± e cos θ): its eccentricity, its type and its directrix. 60 problems with worked solutions; in 60 of them every equation is proved by a computer algebra system.

1 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 8.322

medium

For the conic \( \displaystyle r = \frac{5}{3 \cos{\left(\theta \right)} + 4} \), find the eccentricity, identify the conic, and find the directrix.

Divide top and bottom by the constant in the denominator to get r = ed/(1 ± e cos θ).
Reviewed
\[ \frac{5}{4 \left(\frac{3 \cos{\left(\theta \right)}}{4} + 1\right)} = \frac{5}{3 \cos{\left(\theta \right)} + 4} \]
The standard form.✓ Proved
\[ \frac{\left|{3}\right|}{4} = \frac{3}{4} \]
e = 3/4: < 1, an ellipse.✓ Proved
\[ \frac{5}{4 \cdot \frac{3}{4}} = \frac{5}{3} \]
ed = 5/4, so d = 5/3.✓ Proved

Answer \( e = \frac{3}{4},\ \text{ellipse},\ \text{directrix } x = 5/3 \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More conics in polar form practice

For the conic \( \displaystyle r = \frac{10}{5 \cos{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.310medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{6 \sin{\left(\theta \right)} + 1} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.311medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{3}{5 - 5 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.312medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{11}{5 - 7 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.313medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{4 \sin{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.314medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{5}{1 - 6 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.315medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{5 \sin{\left(\theta \right)} + 4} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.316medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{2}{1 - 5 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.317medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{3 - 2 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.318medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{5}{6 \sin{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.319medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{1}{2 - 3 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.320medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{4}{4 - \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.321medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{7 \cos{\left(\theta \right)} + 1} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.323medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{6 \cos{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.324medium✓ Nihil obstat

For the conic \( \displaystyle r = \frac{10}{1 - 2 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.432medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{4 - 5 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.433medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{1 - 2 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.434medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{5 - 4 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.435medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{3}{5 - 8 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.436medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{5 - 6 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.437medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{2 - 4 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.438medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{4 - 3 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.439medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{4}{3 \sin{\left(\theta \right)} + 4} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.440medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{2 - 2 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.441medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{6 \cos{\left(\theta \right)} + 1} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.442medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{1 - \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.443medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{4}{4 - 8 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.444medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{2 \sin{\left(\theta \right)} + 5} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.445medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{1 - 6 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.446medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{6 \cos{\left(\theta \right)} + 1} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.454medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{8 \sin{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.455medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{\sin{\left(\theta \right)} + 5} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.456medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{6 \sin{\left(\theta \right)} + 5} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.457medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{2}{3 - 7 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.458medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{9}{1 - 5 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.459medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{2 \sin{\left(\theta \right)} + 5} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.460medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{11}{8 \sin{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.461medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{9}{3 - 8 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.462medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{4 - \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.463medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{11}{3 \sin{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.505medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{3}{6 \sin{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.506medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{10}{8 \sin{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.507medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{4 - 6 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.508medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{3 \cos{\left(\theta \right)} + 4} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.509medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{1}{5 \cos{\left(\theta \right)} + 5} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.510medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{11}{4 \sin{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.511medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{9}{3 - 2 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.512medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{4 - 7 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.513medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{2 - 4 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.514medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{8}{6 \cos{\left(\theta \right)} + 4} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.543medium✓ Nihil obstat

For the conic \( \displaystyle r = \frac{1}{6 \cos{\left(\theta \right)} + 2} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.544medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{7}{4 - 3 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.545medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{2}{1 - 7 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.546medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{2 - 4 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.547medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{5 \sin{\left(\theta \right)} + 3} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.548medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{12}{7 \sin{\left(\theta \right)} + 1} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.549medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{6}{1 - \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.550medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{1}{3 - 4 \cos{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.551medium✓ Every equation proved

For the conic \( \displaystyle r = \frac{2}{5 - 8 \sin{\left(\theta \right)}} \), find the eccentricity, identify the conic, and find the directrix.

Problem 8.552medium✓ Every equation proved