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Chapter 6: Problem 29

Find the center and radius of the circle whose equation in polar coordinates is \(r=3 \cos \theta\).

Short Answer

Expert verified
The center of the circle is (3, 0) and the radius is 1.

Step by step solution

01

Convert to Cartesian coordinates

First, we'll convert the given polar equation to Cartesian coordinates. Recall the following relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) \(y = r \sin \theta\) The given equation is \(r = 3\cos \theta\). We can find the Cartesian equation by substituting the expressions for x and y above: \(r = \frac{x}{\cos \theta} = 3\cos \theta\)
02

Simplify the equation

Next, we'll multiply both sides of the equation by \(\cos \theta\) to eliminate the trigonometry: \(x = 3\cos^2 \theta\) Now, recall that \(\cos^2 \theta = 1 - \sin^2 \theta\). Substituting this into the equation, we get: \(x = 3 - 3\sin^2 \theta\) Now, substitute the expression for y back in \(y = r \sin \theta\): \(x = 3 - 3\left(\frac{y}{3}\right)^2\)
03

Identify the center and radius

We've now found the Cartesian equation of the circle: \(x = 3 - y^2\) Notice that this equation has the general form \(x = h - (y-k)^2\) Comparing the two equations, we can identify the center as (h, k) and the radius as r for the circle: Center (h, k) = (3, 0) Radius r = 1 The center of the circle is (3, 0) and the radius is 1.

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