Problem 56 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the rectangular equation. [ Hint: First convert the equation to polar coordinates.] $$x^{2}+y^{2}=\left(x^{2}+y^{2}-x\right)^{2}$$

Step-by-Step Solution

Answer

Graph is two lines through the origin at 45° and 135°.

Step 1: Identify Original Equation

The given equation is \( x^2 + y^2 = (x^2 + y^2 - x)^2 \). We will start by identifying how to convert this into polar coordinates.

Step 2: Convert to Polar Coordinates

Remember the relationships between rectangular and polar coordinates: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Also, \( x^2 + y^2 = r^2 \). Substitute these into the equation.

Step 3: Substitute and Simplify

Substituting, the equation becomes \( r^2 = (r^2 - r\cos(\theta))^2 \). Expand the right side to get \( r^2 = r^4 - 2r^3\cos(\theta) + r^2\cos^2(\theta) \).

Step 4: Simplify Further

Rearrange the terms to have \( r^4 - 2r^3\cos(\theta) + r^2\cos^2(\theta) - r^2 = 0 \). Factor out \( r^2 \) to get \( r^2(r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1) = 0 \).

Step 5: Consider Factor Solutions

The first factor \( r^2 = 0 \) implies \( r = 0 \), which corresponds to the origin. For the second factor, simplify: \( r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1 = 0 \).

Step 6: Complete the Square and Solve for r

To solve \( r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1 = 0 \), rearrange as \( (r - \cos(\theta))^2 = 1 - \cos^2(\theta) \), which gives \( (r - \cos(\theta))^2 = \sin^2(\theta) \).

Step 7: Result and Graphing

Taking the square root, \( r - \cos(\theta) = \pm \sin(\theta) \) gives two solutions: \( r = \cos(\theta) + \sin(\theta) \) and \( r = \cos(\theta) - \sin(\theta) \). Graphing these in polar coordinates results in two intersecting lines through the origin at angles corresponding to \( 45^\circ \) and \( 135^\circ \), intersecting at the origin.

Key Concepts

Graph Sketching

Graph sketching involves drawing a visual representation of equations on a coordinate plane. This exercise transitions from rectangular (or Cartesian) coordinates to polar coordinates. When graphing, it's useful to note the intercepts, symmetry, and shape. In this example, we transform the given equation to a polar form to simplify the graphing process.

• Identify key points such as where the graph crosses the axes.
• Look for symmetry about the origin or the line of interest.
• Consider how the graph behaves as variables tend to infinity or zero.

In this problem, the solution explores the nature of the equation, revealing lines intersecting at the origin, providing a simpler view for sketching.

Rectangular to Polar Conversion

Converting from rectangular to polar coordinates can simplify complex equations. In polar coordinates, each point in the plane is determined by a radius and an angle. The relationships between the two coordinate systems are crucial:

• The x-coordinate translates to polar coordinates via: \( x = r\cos(\theta) \)
• The y-coordinate translates via: \( y = r\sin(\theta) \)
• The equation for a circle or distance from origin is: \( x^2 + y^2 = r^2 \)

Substituting these into the given equation transforms it from a potentially complicated Cartesian form to a more straightforward polar form, making it easier to analyze and graph.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angle values. They are vital when working with polar equations as they allow for simplification and manipulation of expressions. In this solution, we use identities to help rearrange and simplify our polar equation. Key identities used include:

• Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Double Angle Formulas and others can help transform and solve equations

Understanding and using these identities can transform complex problems into more manageable computations, aiding in solving and graphing equations.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square binomial, which can then be easily solved or simplified. This technique is helpful in both algebra and coordinate transformations. In the polar equation derived from the original, you've seen the expression \( r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1 = 0 \).

• By rearranging and completing the square as \( (r - \cos(\theta))^2 = 1 - \cos^2(\theta) \), efficient solutions can be found.
• This helps break down the equation into simple binomials that are easier to manipulate and solve.

Completing the square is particularly useful in finding centers and radii for circles, parabolas, and other conics.