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Chapter 10: Problem 72

Find a polar equation for each conic section. Assume one focus is at the origin.

Short Answer

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Question: Determine the polar equation of a given conic section, assuming one of its foci is at the origin. Answer: To find the polar equation for a specific conic section, first identify the type of conic section (ellipse, parabola, or hyperbola) and determine its parameters (such as \(a\), \(e\), and \(p\)). Then, plug these parameters into the respective general polar equations mentioned in the solution steps above.

Step by step solution

01

Find the general polar equation for an ellipse

The general polar equation for an ellipse with a focus at the origin is given by: \[ r(\theta) = \frac{a(1 - e^2)}{1 - e \cos(\theta)} \] where \(a\) is the semi-major axis, \(e\) is the eccentricity, and \(\theta\) is the angle between the origin and that point.
02

Obtain the polar equation for a specific ellipse

Given the values of \(a\) and \(e\), plug them into the general polar equation in Step 1 to obtain the polar equation for a specific ellipse. For a parabola with a focus at the origin:
03

Find the general polar equation for a parabola

The general polar equation for a parabola with its focus at the origin is given by: \[ r(\theta) = \frac{p}{1 + \cos(\theta)} \] where \(p\) is the distance between the vertex of the parabola and its focus, and \(\theta\) is the angle between the origin and that point.
04

Obtain the polar equation for a specific parabola

Given the value of \(p\), plug it into the general polar equation in Step 3 to obtain the polar equation for a specific parabola. For a hyperbola with a focus at the origin:
05

Find the general polar equation for a hyperbola

The general polar equation for a hyperbola with a focus at the origin is given by: \[ r(\theta) = \frac{a(e^2 - 1)}{1 + e \cos(\theta)} \] where \(a\) is the semi-major axis, \(e\) is the eccentricity, and \(\theta\) is the angle between the origin and that point.
06

Obtain the polar equation for a specific hyperbola

Given the values of \(a\) and \(e\), plug them into the general polar equation in Step 5 to obtain the polar equation for a specific hyperbola. For each conic section, once you have the parameters (such as \(a\), \(e\), and \(p\)), simply plug them into their respective general polar equations mentioned above to find the polar equation for that specific conic section.

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Most popular questions from this chapter

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of an ellipse centered at the origin is \(2 b^{2} / a=2 b \sqrt{1-e^{2}}\)

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices (卤9,0) and eccentricity \(\frac{1}{3}\)

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Prove that the equations $$x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t,$$ where \(a, b, c,\) and \(d\) are real numbers, describe a circle of radius \(R\) provided \(a^{2}+c^{2}=b^{2}+d^{2}=R^{2}\) and \(a b+c d=0.\)

Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at the origin with major and minor axes of lengths 12 and \(2,\) on the \(x\) - and \(y\) -axes, respectively, generated clockwise
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