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Essential Calculus: Early Transcendentals
Found in: Page 535
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Write a polar equation of a conic with the focus at the origin and the given data.

Parabola, vertex \({\rm{(4,3}}\pi {\rm{/2)}}\)

Polar equation of a conic with the focus at the origin and the parabola, vertex \(\left( {{\rm{4,}}\frac{{{\rm{3}}\pi }}{{\rm{2}}}} \right)\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Conic sections are represented by polar equation.

Parabola with vertex \(\left( {{\rm{4,}}\frac{{{\rm{3}}\pi }}{{\rm{2}}}} \right)\).

\(\begin{aligned}{l}{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm ecos\theta }}}},\;\;\\{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 \pm esin\theta }}}}\end{aligned}\)

Step 2: Polar equation.

\(e = 1\)

Pole and vertex \(\left( {{\rm{4,}}\frac{{{\rm{3}}\pi }}{{\rm{2}}}} \right)\)

Vertex is at the same distance from the foci as from the Directrix \({\rm{d = 2 \times 4 = 8}}\)

Polar Equation is,

\(\begin{aligned}{c}{\rm{r = }}\frac{{{\rm{ed}}}}{{{\rm{1 - esin\theta }}}}\\{\rm{ = }}\frac{{\rm{8}}}{{{\rm{1 - sin\theta }}}}\end{aligned}\)

Step 3: Polar equation of a conic with the focus at the origin is.

\({\rm{ = }}\frac{{\rm{8}}}{{{\rm{1 - sin\theta }}}}\)

Most popular questions for Math Textbooks

Sketch the curve with the given polar equation by first sketching the graph of as a function of\({\rm{\theta }}\) in Cartesian coordinates.

\({\rm{r = \theta ,\theta > 0}}\)

Use a calculator to find the length of the curve correct to four decimal places. If necessary, graph the curve to deter-mine the parameter interval \({\rm{r = sin(\theta /4)}}\).

Sketch the curve with the given polar equation by first sketching the graph of \({\rm{r}}\) as a function of \({\rm{\theta }}\) in Cartesian coordinates. \({\rm{r = 3 + 4cos\theta }}\)

Find a polar equation for the curve represented by the given Cartesian equation.

\({\rm{4}}{{\rm{y}}^{\rm{2}}}{\rm{ = x}}\)

Sketch the curve with the given polar equation by first sketching the graph \({\rm{r}}\)as a function of \({\rm{\theta }}\)Cartesian coordinates.

\({\rm{r = 2 + sin\theta }}\)
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