Problem 50 - Algebra and Trigonometry

Question

47–50 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

The graph is a circle of radius 1 centered at the origin overlapped by a cardioid with cusp at the origin.

Step 1: Convert to Polar Coordinates

The given rectangular equation is \(x^2 + y^2 = (x^2 + y^2 - x)^2\). To convert this to polar coordinates, recall the relationships: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) and that \(x^2 + y^2 = r^2\). Substitute these into the equation to get: \(r^2 = (r^2 - x)^2\). Simplifying the substitution, \(x = r \cos(\theta)\), so the equation becomes \(r^2 = (r^2 - r\cos(\theta))^2\).

Step 2: Simplify the Polar Equation

Take the right side expression \((r^2 - r\cos(\theta))^2\) and expand it: \((r^2 - r\cos(\theta))^2 = r^4 - 2r^3\cos(\theta) + r^2\cos^2(\theta)\). Ensure both sides are equal: \(r^2 = r^4 - 2r^3\cos(\theta) + r^2\cos^2(\theta)\). Rearranging yields the polynomial \(r^4 - 2r^3\cos(\theta) + r^2\cos^2(\theta) - r^2 = 0\).

Step 3: Factor the Polynomial Equation

Factor the polynomial expression from Step 2: \(r^2(r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1) = 0\). This factors into two components: \(r^2 = 0\) or \(r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1 = 0\). For \(r^2 = 0\), we have \(r = 0\), which corresponds to a single point at the origin.

Step 4: Solve the Nontrivial Equation

For the second factor, \(r^2 - 2r\cos(\theta) + \cos^2(\theta) - 1 = 0\), recognize it as a quadratic in \(r\). Use the quadratic formula \(r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = -2\cos(\theta)\), and \(c = \cos^2(\theta) - 1\). Solve for \(r\): \(r = \cos(\theta)\) or \(r = 1\), which give polar equations for circles.

Step 5: Sketch the Graph

We have two polar curves: \(r = \cos(\theta)\), which is a cardioid centered on the origin (with its cusp at the origin) and \(r = 1\), which is a circle with radius 1. The graph consists of a circle centered at the origin with radius 1 and a cardioid overlapping the circle at some points. Plot these two curves on the polar coordinate system to visualize the solution.

Key Concepts

Rectangular Equation

A rectangular equation is an expression involving variables, typically in the Cartesian coordinate system with variables like \(x\) and \(y\). In this Cartesian framework, each equation corresponds to a specific curve or line when graphed on a plane. In the given problem, the equation is \(x^2 + y^2 = (x^2 + y^2 - x)^2\).
This type of equation is often used because it directly relates to physical distances and can be easily understood in the context of standard linear graphs. Several mathematical problems involve converting these equations into other forms to make them easier to solve or graph. This brings us to the concept of coordinate conversion, where rectangular equations are expressed in terms of different coordinate systems, such as polar coordinates.

Coordinate Conversion

Coordinate conversion involves changing the description of a point from one coordinate system to another. For example, with a rectangular equation in Cartesian coordinates, you can convert it to polar coordinates to potentially simplify the problem.
Polar coordinates describe a point in the plane using a radius \(r\) and an angle \(\theta\). For conversion, the relationships \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) are used. Additionally, \(x^2 + y^2 = r^2\) gives us a direct polar form equivalence.

• In our example, the conversion transforms \(x^2 + y^2 = (x^2 + y^2 - x)^2\) into \(r^2 = (r^2 - r \cos(\theta))^2\).
• By simplifying this polar form, mathematical manipulation becomes more manageable, and special graph features can be more easily identified, such as symmetry and intercepts.

Graph Sketching

Graph sketching is a skill that involves visualizing the equation's behavior in one or more coordinate systems. After converting to polar coordinates, sketching aids in understanding the geometry of the equation's solution.
A good sketch starts with identifying key features:

• Intercepts, points where the graph intersects axes are critical.
• Symmetry shows how the graph appears unchanged when flipped or rotated.

After identifying these features in the graph, you can draw the general shape it takes. For this problem, sketching involves both a circle \(r = 1\) and a cardioid \(r = \cos(\theta)\). Both shapes should be plotted to see their interactions, noting how they overlap.

Polar Equations

Polar equations use polar coordinates to define curves based on an angle \(\theta\) and radius \(r\). This coordinate system often simplifies problems involving symmetry and radial distance.
The problem highlighted gives two solutions:

• \(r = \cos(\theta)\), representing a cardioid centered at the origin with its cusp touching the origin.
• \(r = 1\), representing a circle with radius 1 centered at the origin.

Understanding these polar equations in terms of their geometric shapes makes solving and sketching them more intuitive. With practice, polar equations expose the symmetry and natural design of the curves, easing the transition from complex algebraic manipulation to visual graph interpretation.