Problem 25 - Calculus Early Transcendentals

Question

\(21-26\) Find a polar equation for the curve represented by the given Cartesian equation. $$x^{2}+y^{2}=2 c x$$

Step-by-Step Solution

Answer

The polar equation is \(r = 2c \cos \theta\).

Step 1: Identify the Cartesian Equation

We are given the Cartesian equation \(x^2 + y^2 = 2cx\). Our goal is to convert this equation into a polar form. In Cartesian coordinates, \(x\) and \(y\) are the variables, while in polar coordinates, \(r\) and \(\theta\) (theta) describe the same points on a plane.

Step 2: Convert Cartesian Coordinates to Polar Coordinates

Recall the conversion formulas from Cartesian to polar coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(x^2 + y^2 = r^2\). Using these substitutions, we will replace \(x\) and \(y\) in the Cartesian equation.

Step 3: Substitute and Simplify

Substitute the polar coordinate expressions into the Cartesian equation:\[x^2 + y^2 = 2cx \\Rightarrow r^2 = 2c(r \cos \theta) \\Rightarrow r^2 = 2cr \cos \theta\]Divide the entire equation by \(r\) assuming \(r eq 0\) (we exclude the origin) to simplify:\[r = 2c \cos \theta\]

Step 4: Polar Equation Result

The polar equation for the curve represented by the given Cartesian equation \(x^2 + y^2 = 2cx\) is \(r = 2c \cos \theta\). This implies that the graph is centered at the pole with variable radius depending on \(\theta\).

Key Concepts

Cartesian coordinates to polar coordinates

Understanding how to switch from Cartesian coordinates to polar coordinates is essential in mathematics. Cartesian coordinates use an ordered pair \(x, y\) to describe the position of a point on a plane. This system is based on two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). In contrast, polar coordinates provide another way to locate a point. They use \(r, \theta\), where \r\ is the distance from the origin (or pole) and \theta\ is the angle from the positive x-axis.

This conversion helps us describe equations and points using angles and radii, which is especially handy in scenarios involving circular or rotational symmetry. For example, in the transformation of the equation given, the Cartesian form describes the shape in a rectangular grid, while the polar form represents it in terms of circles and angles. This perspective can sometimes simplify complex mathematical problems.

Conversion formulas

To convert Cartesian coordinates to polar coordinates, we use specific formulas. Remember, the goal is to express \(x\) and \(y\) using \(r\) and \(\theta\):

- **For x:** \x = r \cos \theta\
- **For y:** \y = r \sin \theta\
- **For x and y:** \x^2 + y^2 = r^2\

These formulas allow us to convert a Cartesian equation into its polar form. In the provided exercise, these formulas were crucial in translating \(x^2 + y^2 = 2cx\) into its polar equivalent.

We substituted \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equation, allowing the Cartesian terms to be rewritten in a way that highlights the relationship between the circle's radius \(r\), angle \(\theta\), and the given constant \(c\).

Polar equations

Polar equations express relationships and shapes in terms of the radius \(r\) and angle \theta\. These equations can provide a clearer and more intuitive representation of shapes involving circles or spirals.

In the exercise example, the Cartesian equation \(x^2 + y^2 = 2cx\) was converted to a polar equation, \(r = 2c \cos \theta\). This equation implies a relationship between the radius and angle, indicating a circle's positioning within the polar coordinate system.

Simplifying equations in this form can reveal the geometric nature of the curve, such as identifying it as a circle. Polar equations transform complex Cartesian expressions into neat forms that highlight symmetry and simplify problem-solving by focusing on radial distance and angular orientation.