Problem 58 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r=2 \cos \theta-4 \sin \theta$$

Step-by-Step Solution

Answer

The Cartesian equation is \((x - 1)^2 + (y + 2)^2 = 5\). It represents a circle with center \((1, -2)\) and radius \(\sqrt{5}\).

Step 1: Recall Polar-Cartesian Conversion

To convert polar coordinates to Cartesian coordinates, we use the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r^2 = x^2 + y^2 \).

Step 2: Replace \( r \cos \theta \text{ and } r \sin \theta \) in Polar Equation

Given the polar equation: \( r = 2 \cos \theta - 4 \sin \theta \), multiply through by \( r \) to replace \(r \cos \theta\) and \(r \sin \theta\). We get: \( r^2 = 2r \cos \theta - 4r \sin \theta \).

Step 3: Substitute with Cartesian Equivalents

Substitute \( r \cos \theta = x \) and \( r \sin \theta = y \), then \( r^2 = x^2 + y^2 \). Thus, the equation becomes \( x^2 + y^2 = 2x - 4y \).

Step 4: Rearrange and Simplify Equation

Move all terms to one side to get \( x^2 + y^2 - 2x + 4y = 0 \).

Step 5: Complete the Square

To rewrite in a more familiar form, complete the square:- For \(x\): \( (x - 1)^2 = x^2 - 2x + 1 \)- For \(y\): \( (y + 2)^2 = y^2 + 4y + 4 \)Thus, the equation becomes \((x - 1)^2 + (y + 2)^2 = 5\).

Step 6: Identify and Sketch the Graph

The equation \((x - 1)^2 + (y + 2)^2 = 5\) is a circle with center \((1, -2)\) and radius \( \sqrt{5} \). Sketch the circle on the Cartesian plane first, and then translate it back to the polar plane where \(r\) is the radial distance and \(\theta\) the angle.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using a radius and an angle. Unlike Cartesian coordinates, which use two perpendicular axes (x and y) to define a point, polar coordinates describe a point in terms of:

• **Radius (r):** the distance from the point to the origin.
• **Angle (\(\theta\)):** the angle formed by a line from the origin to the point and the positive x-axis.

This system is particularly useful in situations where the geometry or symmetry of a problem is radial or circular. In the context of our exercise, converting from polar to Cartesian is essential for manipulating and graphing the given equation.
When dealing with polar equations, the radial coordinate \(r\) can sometimes be negative, implying that the direction is opposite to \(\theta\). However, in many common cases like our example, \(r\) is positive, making graphing straightforward by plotting \(r\) for various angles \(\theta\). By understanding these elements, students can transition from one representation to another, offering deeper insights into mathematical concepts.

Cartesian Coordinates

Cartesian coordinates use two numbers to determine a point in a plane: the x-coordinate and the y-coordinate. These are based on two intersecting perpendicular lines or axes, known as the x-axis and the y-axis. Cartesian coordinates are incredibly intuitive and useful, as they simplify calculations via algebraic expressions.
Converting from polar to Cartesian coordinates involves using the relationships:

• **\(x = r \cos \theta\):** Represents the projection of the point on the x-axis.
• **\(y = r \sin \theta\):** Represents the projection on the y-axis.

To complete the conversion, note that \(r^2 = x^2 + y^2\). This allows us to translate polar equations into the Cartesian plane, making it easier to visualize and analyze, as seen in the step-by-step solution. For the given equation, we used these relationships to move from the polar form \(r = 2 \cos \theta - 4 \sin \theta\) to the Cartesian form \(x^2 + y^2 = 2x - 4y\), eventually transforming it into a circle equation.

Completing the Square

Completing the square is a method used in algebra to transform a quadratic equation into a more recognizable form. This technique is particularly helpful when dealing with circles or conic sections in Cartesian coordinates. For our exercise, this process simplifies and rearranges the equation into a standard form.
To complete the square for our equation \(x^2 + y^2 - 2x + 4y = 0\):

• Identify the quadratic terms separately for \(x\) and \(y\).
• For \(x\): Reorder \(x^2 - 2x\). The square can be completed by adding \(1\) to get \((x - 1)^2\).
• For \(y\): Rearrange \(y^2 + 4y\), then add \(4\) to complete it as \((y + 2)^2\).

After completing the square, we simplify our equation to \((x - 1)^2 + (y + 2)^2 = 5\), which represents a circle centered at \((1, -2)\) with a radius \(\sqrt{5}\). This concise form clearly reveals the graph's shape and location in the Cartesian plane, facilitating both comprehension and visualization for students.