/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 \(1-8=\) Write a polar equation ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 1

\(1-8=\) Write a polar equation of a conic with the focus at the origin and the given data. Ellipse, eccentricity \(\frac{1}{2}, \quad\) directrix \(x=4\)

Short Answer

Expert verified
The polar equation of the ellipse is \( r = \frac{4}{2 - \cos\theta} \).

Step by step solution

01

Understanding the Components

A conic section can be represented in polar coordinates with the focus at the origin using the formula \( r = \frac{ed}{1 - e\cos\theta} \), where \( e \) is the eccentricity and \( d \) is the distance to the directrix. Here, we have an ellipse (since \( e = \frac{1}{2} < 1 \)) and a directrix at \( x = 4 \).
02

Use the Eccentricity

For an ellipse, the eccentricity \( e \) is given as \( \frac{1}{2} \). This value will be used directly in our formula for the polar equation of the conic.
03

Determine Directrix Distance

The distance \( d \) to the directrix from the focus, positioned at the origin, is \( d = 4 \). We will use this value in the formula \( r = \frac{ed}{1 - e\cos\theta} \) with \( e = \frac{1}{2} \) and \( d = 4 \).
04

Substitute into the Polar Equation

Substituting the values we know into the polar equation formula, we have:\[ r = \frac{\left( \frac{1}{2} \right) \times 4}{1 - \frac{1}{2}\cos\theta} \]Simplify it to get:\[ r = \frac{2}{1 - \frac{1}{2}\cos\theta} \].
05

Simplification of the Polar Equation

Multiple both numerator and denominator by 2 to eliminate fractions in the denominator:\[ r = \frac{4}{2 - \cos\theta} \]. This is the polar form of the ellipse with the given conditions.

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Key Concepts

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

Conic Sections
Conic sections are the curves formed by the intersection of a plane with a double-napped cone. These include circles, ellipses, parabolas, and hyperbolas, each having unique properties. They are defined based on the angle at which the plane intersects the cone. For example:
  • If the plane is perpendicular to the axis of the cone, it forms a circle.
  • When the plane cuts through at an angle but doesn鈥檛 pass through the base, it forms an ellipse.
  • Cutting parallel to one of the sides of the cone results in a parabola.
  • If the plane cuts through both nappes, a hyperbola is formed.
Conics are fundamental in geometry and frequently appear in various scientific contexts, such as planetary orbits in astronomy.
Eccentricity
Eccentricity is a critical concept in understanding conic sections. It is a measure of how much a conic section deviates from being circular. The eccentricity (\(e\)) of each conic:
  • For a circle, \(e = 0\)
  • The ellipse has \(0 < e < 1\)
  • A parabola always has \(e = 1\)
  • For a hyperbola, \(e > 1\)
In the provided exercise, we deal with an ellipse, hence \(e = \frac{1}{2}\). This value indicates that the shape is more circular but still elliptical. Eccentricity also has implications for the speed variation of objects moving along these paths due to gravitational influences.Eccentricity helps us to understand not just the shape, but also some of the physics behind movements along these curves.
Directrix
The directrix is a fixed reference line used to define and describe a conic section. It works together with eccentricity to define the path of the conic.In polar coordinates, the distance to the directrix from the focus plays a critical role.
  • For ellipses and hyperbolas, it is used to define the shape involving both the directrix and eccentricity.
  • For parabolas, the focus and directrix work together to define the parabola, with every point equidistant from the focus and the directrix.
In our exercise, the directrix is given by the equation \(x = 4\), which helps form the polar equation of the ellipse using the formula:\[r = \frac{ed}{1 - e\cos\theta}\]where \(d\) represents the distance to the directrix, and for the ellipse, \(d = 4\). By understanding the role of the directrix, students can better grasp how conic sections are formed and manipulated within different coordinate systems.

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Most popular questions from this chapter

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. $$r=4 \sin 3 \theta$$

Sketch the curve with the given polar equation by first sketching the graph of \(r\) as a function of \(\theta\) in Cartesian coordinates. $$r^{2} \theta=1$$

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$0 \leqslant r<2, \quad \pi \leqslant \theta \leqslant 3 \pi / 2$$

Find the exact length of the curve. $$x=1+3 t^{2}, \quad y=4+2 t^{3}, \quad 0 \leqslant t \leqslant 1$$

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $$x=t+\sqrt{t}, \quad y=t-\sqrt{t}, \quad 0 \leqslant t \leqslant 1$$
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