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Chapter 8: Problem 186

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented. \(r=\frac{6}{\cos \theta+3 \sin \theta}\)

Short Answer

Expert verified
The Cartesian equation is \( x + 3y = 6 \), which is a line.

Step by step solution

01

Recognize Polar to Cartesian Conversion Formulas

To convert from polar to Cartesian coordinates, we use the relationships: - \( x = r \cos \theta \) - \( y = r \sin \theta \)Additionally, we have \( r = \sqrt{x^2 + y^2} \).
02

Multiply by the Denominator

Multiply both sides of the polar equation by the denominator \( \cos \theta + 3 \sin \theta \) to clear the fraction:\[ r(\cos \theta + 3 \sin \theta) = 6 \]
03

Use Polar-Cartesian Relationships

Substitute the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation:\[ x + 3y = 6 \]
04

Identify the Conic Section

The equation \( x + 3y = 6 \) is in the form of \( Ax + By = C \), which represents a line. A line is a degenerate form of a conic section.

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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 fascinating shapes that arise when we slice a cone with a plane. These sections include circles, ellipses, parabolas, and hyperbolas. When the plane intersects the cone in different angles and positions, it will produce different types of curves.
  • A **circle**, for example, is obtained when the slicing plane is parallel to the base of the cone.
  • An **ellipse** can be found if the slicing plane is tilted, but not parallel to the side of the cone.
  • A **parabola** emerges when the plane is parallel to the slant of the cone.
  • Finally, a **hyperbola** results when the plane cuts through both halves of the cone.
While a line isn鈥檛 a typical conic section, it can be considered a degenerate conic. This happens when the slicing plane just grazes the cone, not cutting through it deeply. Therefore, in our exercise, the equation forming a line is a degenerate case, not falling into the main categories of conic sections.
Cartesian Coordinates
Cartesian coordinates are used to describe the location of a point within a plane. Named after the mathematician Ren茅 Descartes, this system uses two axes, usually denoted as the x-axis and y-axis, which are perpendicular to each other.
  • The horizontal axis (x-axis) and the vertical axis (y-axis) divide the plane into four quadrants.
  • Any point can be represented as an ordered pair \(x, y\), where 'x' shows the position along the x-axis and 'y' along the y-axis.
  • This system enables easy graphing of equations and easy determination of the position of points.
Using Cartesian coordinates facilitates the conversion from polar equations, like in our exercise, into a standard linear format. By substituting relations like \(x = r \cos \theta\) and \(y = r \sin \theta\), we map the problem into simpler, more familiar territory.
Polar Equations
Polar equations involve a distinct way of expressing points within a plane using radius and angles. In this coordinate system, each point is determined by a radius 'r' from the origin (pole) and an angle \(\theta\) from a reference direction, usually the positive x-axis.
  • This system is specifically helpful when dealing with curves and equations related to circular and rotational motion.
  • The key to converting from polar to Cartesian equations lies in the relationships:
    • \( x = r \cos \theta \)
    • \( y = r \sin \theta \)
    • \( r = \sqrt{x^2 + y^2} \)
For the given polar equation \(r=\frac{6}{\cos \theta+3 \sin \theta}\), understanding these relationships allows us to convert and simplify it into a Cartesian linear form. This transformation enabled us to recognize the type of conic section represented by the equation.

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Most popular questions from this chapter

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