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Chapter 10: Problem 36

Finding a Polar Equation In Exercises \(33-38\) , find a polar equation for the conic with its focus at the pole and the given eccentricity and directrix. (For convenience, the equation for the directrix is given in rectangular form.) $$\begin{array}{ll}{\text { Conic }} & {\text { Eccentricity }} \\ {\text { Ellipse }} & {\quad e=\frac{5}{6}}\end{array} \begin{array}{l}{\text { Directrix }} \\ {y=-2}\end{array}$$

Short Answer

Expert verified
The polar equation of the given ellipse is \(r = \frac{2}{1 - \frac{5}{6}\cos(\theta)}\).

Step by step solution

01

Recall the Standard Polar Equation of Conic Sections

The standard polar equation of a conic section with the focus at the pole is given by \[r=\frac{p}{1-e\cos(\theta)}\] for 0<e<1 or eccentricity e is between 0 and 1. The constant p is the distance of the directrix line to the origin.
02

Find the Constant p

Given that the directrix line is \(y = -2\), and because the directrix is horizontal, the distance from the origin will simply be the absolute value of -2, which is 2. So p is equal to 2.
03

Plug in the Given Constants

Now, substitute the given eccentricity e = 5/6 and p = 2 into the standard equation. The polar equation of the ellipse is therefore given by \[r=\frac{2}{1-\frac{5}{6}\cos(\theta)}\]

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Key Concepts

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

Eccentricity of Conic Sections
The concept of eccentricity is central to understanding the shape and properties of conic sections. Conic sections, which include circles, ellipses, parabolas, and hyperbolas, are curves obtained by intersecting a plane with a cone. Eccentricity, denoted as 'e', is a measure that describes how much a conic section deviates from being circular.

Eccentricity Values:
  • For a circle, e=0, which means it has no eccentricity since it is perfectly round.
  • For an ellipse, 0
  • A parabola has an eccentricity of e=1. This defines its open and symmetric shape along a single axis.
  • For a hyperbola, e>1, indicating a shape that opens more widely than a parabola.
In the provided exercise, the elliptical conic section has an eccentricity of e=5/6, which means it is fairly elongated, but still retains the characteristic closed curve of an ellipse. The eccentricity provides essential information when forming the polar equation of a conic section.
Polar Coordinates
Polar coordinates offer an alternative way to represent points on a plane, using a radius and angle rather than x and y coordinates. In polar coordinates, a point's location on a plane is determined by:
  • The radius 'r': the straight-line distance from the point to the origin, also known as the pole.
  • The angle '胃' (theta): the angle measured in radians or degrees between the positive x-axis (referred to as the polar axis) and the line from the origin to the point.
Converting to Polar Coordinates:
To find the polar coordinates of a point with rectangular coordinates (x, y), one can use the relationships:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \arctan(\frac{y}{x})\), considering the quadrant where the point is located
Polar equations, such as those for conic sections, often appear simpler and more symmetrical than their rectangular counterparts. This can be particularly useful when dealing with problems that have a clear central point or symmetry, like those involving orbiting bodies or wave patterns.
Calculus
Calculus, a branch of mathematics, provides tools for analyzing changes in physical quantities through its two main subfields: differential calculus and integral calculus. Differential calculus focuses on the rate of change, known as derivatives, while integral calculus deals with accumulation, known as integrals.

Applications in Conic Sections:
  • In the context of conic sections, calculus can be used to determine areas, arc lengths, and the curvature of these shapes.
  • Differential calculus helps analyze the slope of tangent lines to the curves of conics, describing how they change at any given point.
  • Integral calculus can calculate the area enclosed by a conic section or the area between a conic section and a line or another curve.
Calculus also plays a significant role in more complex problems that involve conic sections, such as those in physics, engineering, and astronomy. By facilitating the computation of orbits or trajectories, which are described by conic sections, it can help predict the motion of planets or the path of a projectile.

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Most popular questions from this chapter

Finding the Arc Length of a Polar Curve In Exercises \(59-64\) , use a graphing utility to graph the polar equation over the given interval. Use the integration capabilities of the graphing utility to approximate the length of the curve. $$r=e^{\theta}, \quad[0, \pi]$$

Explorer 18 On November \(27,1963,\) the United States launched Explorer \(18 .\) Its low and high points above the surface of Earth were approximately 119 miles and \(123,000\) miles (see figure). The center of Earth is a focus of the orbit. Find the polar equation for the orbit and find the distance between the surface of Earth and the satellite when \(\theta=60^{\circ} .\) (Assume that the radius of Earth is 4000 miles.)

Finding the Area of a Polar Region Between Two Curves In Exercises \(37-44\) , use a graphing utility to graph the polar equations. Find the area of the given region analytically. Common interior of \(r=3-2 \sin \theta\) and \(r=-3+2 \sin \theta\)

Finding an Angle In Exercises \(107-112,\) use the result of Exercise 106 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\) . Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta .\) Identify the angle \(\psi\) . $$r=3(1-\cos \theta) \quad \theta=\frac{3 \pi}{4}$$

Finding the Standard Equation of a Hyperbola In Exercises \(41-48,\) find the standard form of the equation of the hyperbola with the given characteristics. $$\begin{aligned} \text { Vertices: }(0,2) &,(6,2) \\ \text { Asyruplotes: } y &=\frac{2}{7} x \\ y &=4-\frac{2}{3} x \end{aligned}$$
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