Problem 58 - Thomas Calculus

Question

Find polar equations for the circles in Exercises \(57-64 .\) Sketch each circle in the coordinate plane and label it with both its Cartesian and polar equations. $$(x+2)^{2}+y^{2}=4$$

Step-by-Step Solution

Answer

The polar equation of the circle is \(r = -4 \cos \theta\).

Step 1: Identify the standard form of the circle in Cartesian coordinates

The given equation \((x+2)^2 + y^2 = 4\) is in the standard form of a circle, \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Here, \(h = -2\), \(k = 0\), and \(r = 2\). So, the circle's center is \((-2, 0)\) and its radius is \(2\).

Step 2: Determine the polar coordinates

In polar coordinates, any point \((x, y)\) has coordinates \((r \cos \theta, r \sin \theta)\). Thus, \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 3: Substitute Cartesian coordinates to polar form

Replace \(x\) and \(y\) in the Cartesian equation with their polar equivalents: \((r \cos \theta + 2)^2 + (r \sin \theta)^2 = 4\).

Step 4: Simplify the polar equation

Expand the equation: \((r \cos \theta + 2)^2 = (r^2 \cos^2 \theta + 4r \cos \theta + 4)\). Then add \(r^2 \sin^2 \theta\) from the \(y^2\) term to get \(r^2 \cos^2 \theta + r^2 \sin^2 \theta + 4r \cos \theta + 4 = 4\).

Step 5: Use Pythagorean identity

Apply the identity \(\cos^2 \theta + \sin^2 \theta = 1\) to simplify the equation to \(r^2 + 4r \cos \theta + 4 = 4\).

Step 6: Solve for r in terms of θ

Subtract 4 from both sides: \(r^2 + 4r \cos \theta = 0\). Factor the remaining expression: \(r(r + 4 \cos \theta) = 0\). Thus, \(r = 0\) or \(r = -4 \cos \theta\). Only the latter solution represents the circle beyond the origin.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique and fascinating way to represent points on a plane. Unlike Cartesian coordinates, which use a grid-like system of horizontal and vertical axes defined by \(x\) and \(y\), polar coordinates rely on a central point known as the pole (similar to the origin in Cartesian coordinates). In this system, each point is represented by a distance \(r\) from the pole and an angle \(\theta\) from the positive x-axis. This allows us to express any point in the form \( (r, \theta) \).

• \(r\) is the radius or distance from the pole.
• \(\theta\) is the angle measured in radians usually, although degrees can also be used.

Polar coordinates are particularly useful for dealing with curves and shapes that have symmetry about a point, such as circles and spirals. Converting between Cartesian and polar coordinates is a skill that can simplify complex mathematics involving rotational symmetry.

Cartesian Coordinates

Cartesian coordinates are the familiar grid system we use to map points on a plane. Named after René Descartes, this system uses a horizontal (x) axis and a vertical (y) axis to locate points. Each point is denoted by \( (x, y) \), where \(x\) represents a point's position along the horizontal axis, and \(y\) indicates its position along the vertical axis.

• The Cartesian system is ideal for linear equations and shapes like lines and rectangles.
• The origin is the point \( (0, 0) \), where the two axes intersect.

One of the strengths of Cartesian coordinates is their ability to work seamlessly with algebraic operations, enabling formulas and equations involving straight lines and right angles to be written and solved more easily.

Equation of a Circle

An equation of a circle gives us a concise way to describe all the points that make up the circle. In the Cartesian system, the standard form of a circle's equation is \( (x-h)^2 + (y-k)^2 = r^2 \), where:

• \( (h, k) \) is the center of the circle.
• \(r\) is the radius of the circle.

This equation tells us that every point \( (x, y) \) on the circle is exactly \(r\) units away from the center \( (h, k) \). For the given exercise, \( (x+2)^2 + y^2 = 4 \), the circle's center is found at \( (-2, 0) \) with a radius of 2. The equation helps in identifying circles by simply recognizing this pattern and correctly determining the respective center and radius.

Conversion Between Coordinate Systems

Converting between polar and Cartesian coordinates is a vital skill in mathematics, especially when dealing with problems that involve rotational symmetry or require simpler equations. To convert from Cartesian to polar coordinates, the following relationships are essential:

• \( x = r \cos \theta \) and \( y = r \sin \theta \) help us find the Cartesian coordinates from polar coordinates.
• The formula \( r = \sqrt{x^2 + y^2} \) calculates the distance from the origin to any point.
• \( \theta = \text{atan2}(y, x) \) determines the angle formed with the x-axis.

In the case of circle equations, we often seek to express Cartesian equations in polar form to capture symmetry or simplify calculations. The process may seem complicated initially, but breaking it down step by step—substituting for \(x\) and \(y\), expanding, and applying the Pythagorean identity—streamlines conversion.