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Chapter 11: Problem 10

Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity \(e,\) and a directrix \(x=d,\) where \(d>0\)

Short Answer

Expert verified
Answer: The polar equation of the conic section is given by \(r = e |r\cos{\theta} - d|\).

Step by step solution

01

Express the distance between a point in the conic section and the directrix

Using the Cartesian coordinates, the directrix is a vertical line given by x=d. Let a point P in the conic section be represented by \((x,y)\) or \((r,\theta)\) in Cartesian and polar coordinates, respectively. Then the distance between the directrix and P is given by \(|x-d|\). We need to express this distance in polar coordinates. Since \(x = r\cos{\theta}\), we can rewrite the distance as \(|r\cos{\theta} - d|\).
02

Express the distance between a point in the conic section and the focus

The focus of the conic section is at the origin, and we need to express the distance between a point P in the conic section and the origin in polar coordinates. This distance is simply the radial distance r.
03

Relate the distances using the definition of eccentricity

The definition of the eccentricity e of a conic section is the ratio of the distance between a point P and the focus to the distance between P and the directrix. In our case, this is given by: \[e = \frac{r}{|r\cos{\theta} - d|}\]
04

Isolate r to express the polar equation

To find the polar equation of the conic section, we need to isolate r from the equation above. We can do this by multiplying both sides by \(|r\cos{\theta} - d|\) and then rearranging the terms: \[r = e |r\cos{\theta} - d|\] This is the polar equation of the conic section with a focus at the origin, eccentricity e, and directrix x=d.

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Most popular questions from this chapter

Water flows in a shallow semicircular channel with inner and outer radii of \(1 \mathrm{m}\) and \(2 \mathrm{m}\) (see figure). At a point \(P(r, \theta)\) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on \(r\), the distance from the center of the semicircles. a. Express the region formed by the channel as a set in polar coordinates. b. Express the inflow and outflow regions of the channel as sets in polar coordinates. c. Suppose the tangential velocity of the water in \(\mathrm{m} / \mathrm{s}\) is given by \(v(r)=10 r,\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.5, \frac{\pi}{4}\right)\) or \(\left(1.2, \frac{3 \pi}{4}\right) ?\) Explain. d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r},\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.8, \frac{\pi}{6}\right)\) or \(\left(1.3, \frac{2 \pi}{3}\right) ?\) Explain. e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_{0}\) ) is proportional to \(\int_{1}^{2} v(r) d r .\) Is the total flow through the channel greater for the flow in part (c) or (d)?

Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. $$r=\frac{4}{1+\cos \theta}$$

Find an equation of the line tangent to the hyperbola \(x^{2} / a^{2}-y^{2} / b^{2}=1\) at the point \(\left(x_{0}, y_{0}\right)\)

Sketch the graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work. $$\frac{y^{2}}{16}-\frac{x^{2}}{9}=1$$

Sketch the graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. $$12 x^{2}+5 y^{2}=60$$
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