Problem 41 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r^{2}=-4 r \cos \theta $$

Step-by-Step Solution

Answer

The graph is a circle with center (-2, 0) and radius 2.

Step 1: Understand the Polar Equation

The given polar equation is \( r^2 = -4r \cos \theta \). Our goal is to convert this to a Cartesian form and identify the graph. We know the relationships between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Apply Conversion Formulas

Using the conversion formulas, express \( r \cos \theta \) as \( x \). Substitute \( x \) into the equation to yield: \[ r^2 = -4x \].

Step 3: Replace \( r^2 \) with Cartesian Variables

Recall that \( r^2 = x^2 + y^2 \) in Cartesian coordinates. Substitute \( r^2 \) with \( x^2 + y^2 \) to form the equation: \[ x^2 + y^2 = -4x \].

Step 4: Rearrange for Completion of Square

Move \( -4x \) to the left side to obtain \( x^2 + y^2 + 4x = 0 \). To complete the square for the \( x \)-terms, rewrite \( x^2 + 4x \) as \((x+2)^2 - 4\). So the equation becomes: \( (x+2)^2 - 4 + y^2 = 0 \).

Step 5: Complete the Square and Simplify

Simplify the equation by adding 4 to both sides to remove the negative term: \((x+2)^2 + y^2 = 4\). This is the equation of a circle with center \((-2, 0)\) and radius 2.

Step 6: Identify the Graph

Based on the equation \((x+2)^2 + y^2 = 4\), the graph is a circle centered at \((-2, 0)\) with a radius of 2.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way to describe the position of a point on a plane. Instead of using the traditional Cartesian coordinate system, which uses a pair of numbers to specify a point's location in terms of horizontal and vertical distances from an origin, polar coordinates utilize a different approach.

• The first coordinate, often noted as \( r \), represents the radial distance from the origin (the pole).
• The second coordinate, \( \theta \), indicates the angle from the positive x-axis in a counter-clockwise direction.

The mathematical relationships between polar coordinates \((r, \theta)\) and Cartesian coordinates \((x, y)\) are defined as:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These relationships allow us to convert equations and understand their graphical representations in Cartesian terms. The problem involves converting the polar equation \( r^2 = -4r \cos \theta \) into Cartesian form to better understand its graph.

Cartesian Coordinates

The Cartesian coordinate system is a widely used framework for describing the positions of points on a plane. It utilizes two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each ordered pair \((x, y)\) represents a unique point in this two-dimensional space.

In order to convert from polar coordinates to Cartesian coordinates, use substitutions based on trigonometric identities:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

By substituting these expressions into polar equations, the equations can be re-written in terms of \( x \) and \( y \), like transforming \( r^2 = -4r \cos \theta \) into \( x^2 + y^2 = -4x \). This transformation helps reveal geometrical features, like shapes and positions, in a more familiar Cartesian plane view.

Completing the Square

Completing the square is an algebraic technique often used to simplify equations, revealing hidden geometrical properties such as the center and radius of a circle.It involves rewriting a quadratic equation in a way that makes it easy to identify squares and solve or simplify further.

In the equation \( x^2 + y^2 + 4x = 0 \), completing the square focuses on the "\( x^2 + 4x \)" terms:

• Add and subtract the square of half the coefficient of \( x \), which is \( 2^2 = 4 \).
• This transforms \( x^2 + 4x \) into \( (x+2)^2 - 4 \).

This creates an equation that can be reorganized into a standard circle form.Applying this method reveals the complete equation of the circle: \( (x+2)^2 + y^2 = 4 \). This step is crucial because it makes the circle's properties readily identifiable.

Equation of a Circle

The equation of a circle in standard form provides clear insights about its geometry:

\( (x-h)^2 + (y-k)^2 = r^2 \)

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

From the problem and using completing the square, we derived \( (x+2)^2 + y^2 = 4 \), which uses the standard equation form for circles.

• The term \( (x+2)^2 \) indicates a horizontal shift, moving the circle’s center to \( (-2, 0) \).
• \( y^2 \) indicates that there isn't any vertical shifting, keeping the center at zero on the y-axis.
• The radius, \( r \), comes from \( r^2 = 4 \), thus \( r = 2 \).

Understanding this form reveals the complete nature of the circle, giving a visual representation of the position and size described by the equation.