Problem 81 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(x^2+y^2-2ax=0\)

Step-by-Step Solution

Answer

The polar form of the given equation is \(r=2a\cos\theta\).

Step 1: Replace the variables

The first step is replacing \(x\) and \(y\) with their polar equivalents \(r\cos\theta\) and \(r\sin\theta\), respectively. The given equation thus becomes \(r^2\cos^2\theta + r^2\sin^2\theta - 2ar\cos\theta = 0\)

Step 2: Simplify the equation

Use the Pythagorean identity \(\cos^2\theta + \sin^2\theta = 1\), simplify the equation \(r^2 - 2ar\cos\theta = 0\).

Step 3: Factor out r

Next, we'll take \(r\) out of the expression, giving \(r(r-2a\cos\theta)=0\).

Step 4: Provide the solution in polar form

Finally, it follows from this expression that either \(r=0\) or \(r=2a\cos\theta\). The assumption stated that \(a>0\), Thus, the equation in polar form is \(r=2a\cos\theta\).

Key Concepts

Rectangular to Polar Conversion

In coordinate geometry, we often face the task of changing the form of equations. One common conversion is from rectangular (Cartesian) coordinates to polar coordinates. Let's break this down to understand it better.

Rectangular coordinates are expressed in terms of \(x\) and \(y\), while polar coordinates use the radius \(r\) and angle \(\theta\). The conversion between these systems is achieved using the following formulas:

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

These definitions allow us to rewrite any equation involving \(x\) and \(y\) in terms of \(r\) and \(\theta\). It's like translating a sentence from one language to another, ensuring that the meaning remains unchanged.

In the given exercise, by substituting \(x = r\cos\theta\) and \(y = r\sin\theta\), we convert the rectangular equation \(x^2 + y^2 - 2ax = 0\) into a new expression using polar terms, which becomes the starting point for further simplification. This is the first key step in the task of transforming coordinate systems.

Pythagorean Identity

The Pythagorean Identity is a fundamental mathematical concept that comes in very handy for simplifying trigonometric equations, especially when converting coordinate systems. Let's explore it a bit more.

The Pythagorean Identity states that for any angle \(\theta\):

• \(\cos^2\theta + \sin^2\theta = 1\)

This identity is crucial when dealing with polar conversions. In our exercise, after substituting the polar coordinates, we find terms \(r^2\cos^2\theta + r^2\sin^2\theta\). By recognizing the identity, we can substitute \(\cos^2\theta + \sin^2\theta\) with 1, simplifying our equation greatly.

This step simplifies the expression to \(r^2 - 2ar\cos\theta = 0\). Using the Pythagorean Identity effectively reduces complex terms, making the equation much easier to handle and paving the way for solving it analytically.

Equation Transformation

Transforming equations is akin to sculpting; we shape them to reveal their deeper structure or simplify their expressions. This is a common task when working between coordinate systems.

In the context of our exercise, once the equation \(r^2 - 2ar\cos\theta = 0\) is simplified using the Pythagorean Identity, we need to bring it to polar form explicitly.

The next step is factoring the equation, which means identifying common factors and simplifying the expression. We extract the greatest common factor \(r\), giving us \(r(r - 2a\cos\theta) = 0\). This expression informs us that either \(r = 0\) or \(r = 2a\cos\theta\).

• If \(r = 0\), it corresponds to the origin in the polar plane, which is typically not the focus in polar transformations.
• If \(r = 2a\cos\theta\), we have the desired polar equation.

Hence, the rectangular equation transforms into its polar equivalent, revealing its geometry relative to the origin and angle \(\theta\). This transformation is a culmination of using trigonometric identities and algebraic manipulations.