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Chapter 10: Problem 26

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix. $$e=1, \quad r \cos \theta=5$$

Short Answer

Expert verified
The polar equation is \( r = \frac{5}{1 - \cos \theta} \).

Step by step solution

01

Identify the type of conic

Since the eccentricity \(e\) is equal to 1, the conic is a parabola. A parabola has its focus at the pole and its directrix is at a fixed distance from the pole.
02

Write the directrix in polar coordinates

The given equation for the directrix is \(r \cos \theta = 5\). This represents a vertical line located at \(x = 5\) in rectangular coordinates.
03

Recall the polar equation for a parabola

For a parabola, the polar equation is given by \(r = \frac{ep}{1 + e \cos \theta}\) or \(r = \frac{ep}{1 - e \cos \theta}\), depending on the position of the directrix relative to the pole. Since the directrix is \(r \cos \theta = 5\), this corresponds to \(r = \frac{ep}{1 - e \cos \theta}\).
04

Determine the constant p

Since the eccentricity \(e\) is 1, this simplifies the equation to \(r = \frac{p}{1 - \cos \theta}\). The distance \(p\) is the perpendicular distance from the focus to the directrix, which here is 5.
05

Write the final polar equation

Substitute the value of \(p = 5\) and \(e = 1\) into the equation \(r = \frac{p}{1 - \cos \theta}\) to get the final polar equation of the parabola: \(r = \frac{5}{1 - \cos \theta}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates offer a unique way to express the position of a point in a plane. Unlike the rectangular coordinate system that uses x and y values, polar coordinates determine a point by a distance and an angle. The pair \(r, \theta\) describes a point in this system, where \(r\) represents the radial distance from the origin (or pole) and \(\theta\) represents the angle from the positive x-axis.
  • The angle \(\theta\) can be positive or negative, affecting the direction of the point relative to the pole.
  • When \(r\) is positive, the point is in the direction of \(\theta\), but when \(r\) is negative, the point is considered in the opposite direction.
  • This system is particularly useful when dealing with problems involving angles or circular/spiral patterns, as it simplifies calculations.
Using polar coordinates is essential in expressing the equations of conic sections, such as parabolas, where the conics have a certain symmetry about a point.
Eccentricity
Eccentricity, denoted as \(e\), is a number that characterizes how much a conic section deviates from being circular. It plays a major role in determining the type of conic section in geometry. In simple terms, the eccentricity defines the shape of the conic:
  • If \(e = 0\), the conic is a circle.
  • If \(0 < e < 1\), the figure is an ellipse.
  • An eccentricity of exactly 1, as in our exercise, defines a parabola.
  • If \(e > 1\), the conic is a hyperbola.
Eccentricity influences the equation of the conic in polar coordinates. For a parabola specifically, having \(e = 1\) simplifies the equation, as you see in the polar equation \(r = \frac{p}{1 - \cos \theta}\), making it comparatively straightforward to handle algebraically.
Parabola
A parabola is a distinct conic section with several key characteristics based on its eccentricity and geometric properties. It appears in the context of reflective properties and projectile motion as a U-shaped curve.
  • The parabola has the property where all points are equidistant from a specific point called the focus, and a line known as the directrix.
  • For transformations and analyzing motions, parabolas are extremely relevant.
In polar equations, a parabola can be represented using the formula \(r = \frac{p}{1 - \cos \theta}\), when the directrix is a vertical line, like in our original problem. Here, the position of the directrix influences the form of the equation - making it fundamental in deriving accurate polar equations.
Focus and Directrix
The focus and directrix are critical components in defining a conic section. Specifically, they help determine the parabolic shape and position in space.
  • The focus is a fixed point, and in polar coordinates, it is located at the pole.
  • The directrix is a fixed line, often found in rectangular form, which can be converted into polar coordinates.
In the exercise, we have the focus at the pole and the directrix given by \(r \cos \theta = 5\), indicating its location at the line \(x = 5\). This ensures every point on the parabola remains equidistant from the focus and the directrix, with eccentricity confirming its parabolic nature. Utilizing these components allows us to derive the polar equation accurately.

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Most popular questions from this chapter

Kepler's first law asserts that planets travel in elliptical orbits with the sun at one focus. To find an equation of an orbit, place the pole \(O\) at the center of the sun and the polar axis along the major axis of the ellipse (see the figure). (a) Show that an equation of the orbit is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis. (b) The perihelion distance \(r_{\text {per }}\) and aphelion distance \(r_{\text {aph }}\) are defined as the minimum and maximum distances, respectively, of a planet from the sun. Show that \(r_{\text {per }}=a(1-e) \quad\) and \(\quad r_{\text {aph }}=a(1+e)\) (IMAGE CAN NOT COPY)

Exer. \(69-72\) : Graph the ellipses on the same coordinate plane, and estimate their points of intersection. $$\frac{x^{2}}{3.1}+\frac{(y-0.2)^{2}}{2.8}=1 ; \quad \frac{(x+0.23)^{2}}{1.8}+\frac{y^{2}}{4.2}=1$$

Graph the hyperbolas on the same coordinate plane, and estimate their first- quadrant point of intersection.. $$\frac{(x-0.1)^{2}}{0.12}-\frac{y^{2}}{0.1}=1 ; \frac{x^{2}}{0.9}-\frac{(y-0.3)^{2}}{2.1}=1$$

Sketch the graph of the polar equation. $$r=1-\csc \theta$$

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation $$r=\frac{r_{p e r}(1+e)}{1-e \cos \theta}$$ where \(e\) is the eccentricity of the conic and \(r_{\mathrm{per}}\) is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has \(r_{\text {per }}=9.006\) and \(e=0.056\) Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle. Comet 1959 III \(\quad r_{\text {per }}=1.251, \quad e=1.003\) $$[-18,18,3] \text { by }[-12,12,3]$$
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