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Chapter 8: Problem 57

Convert each equation to polar coordinates and then sketch the graph.\(x^{2}+y^{2}=6 x\)

Short Answer

Expert verified
The polar form is \(r = 6\cos\theta\), representing a circle with center \((3, 0)\) and radius 3.

Step by step solution

01

Identify the Cartesian Equation

We are given the equation in Cartesian coordinates: \(x^2 + y^2 = 6x\). Our goal is to convert this equation into polar coordinates and sketch the graph.
02

Convert to Polar Coordinates

In polar coordinates, the relationships are: \(x = r\cos\theta\) and \(y = r\sin\theta\). Therefore, \(x^2 + y^2 = r^2\). Substitute these into the equation: \(r^2 = 6(r\cos\theta)\).
03

Simplify the Polar Equation

Rewrite the equation as \(r^2 = 6r\cos\theta\). Divide both sides by \(r\) (assuming \(r eq 0\)) to simplify: \(r = 6\cos\theta\).
04

Identify the Equation Type

The equation \(r = 6\cos\theta\) is a standard polar equation representing a circle. It has the form \(r = a\pm b\cos\theta\) or \(r = a\pm b\sin\theta\) indicating a circle.
05

Determine the Circle's Properties

In the equation \(r = 6\cos\theta\), the circle is centered at \((3, 0)\) in Cartesian coordinates with a radius of 3. The center is found as half of the coefficient of \(\cos\theta\), and the radius is this same half value.
06

Sketch the Graph

Draw the circle centered at \((3, 0)\) with a radius of 3. This completes the conversion and graph sketching of the original equation in polar coordinates.

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Key Concepts

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

Cartesian Coordinates
Cartesian coordinates are a system that uses two or three numbers to specify the location of a point in a plane or space. These numbers are coordinates, typically represented as \(x\) and \(y\) in a two-dimensional space. \(x\) indicates the horizontal position from the origin, while \(y\) represents the vertical position.

To understand this system, think of a graph you often see in algebra with an x-axis (horizontal) and a y-axis (vertical). Every point on this graph can be described using an ordered pair \((x, y)\), which tells us how far along each axis the point is located.
  • The origin is represented as \((0, 0)\).
  • If you move right, \ x \ values increase, and if you move left, they decrease.
  • Moving up increases \ y \ values, and moving down decreases them.
This system is fundamental for graphing equations and visualizing relationships between variables.
Polar Equation
Polar coordinates describe the position of a point based on its distance from a reference point and angle from a reference direction. In polar systems, a point is defined by \( r \) and \( \theta \).

The reference point is the pole, similar to the origin in Cartesian coordinates, and the reference direction is the polar axis, similar to the x-axis. \( r \) is the radial distance from the pole to the point, and \( \theta \) is the angle from the positive direction of the polar axis.
  • \

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Most popular questions from this chapter

Graph each equation using your graphing calculator in polar mode.\(r=3 \sin 2 \theta+2 \cos \theta\)

Sketch the graph of each equation by making a table using values of \(\theta\) that are multiples of \(45^{\circ}\).\(r=4 \sin \theta\)

Sketch the graph of each equation by making a table using values of \(\theta\) that are multiples of \(45^{\circ}\).\(r=6 \cos \theta\)

Find the following quotients. Write all answers in standard form for complex numbers. $$\frac{2+3 i}{2-3 i}$$

Graph each equation.\(r=2+2 \cos \theta\)
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