Problem 75 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r^{2}=4 \cos 2 \theta $$

Step-by-Step Solution

Answer

The polar equation \(r^{2}=4 \cos 2 \theta\) represents a limaçon oriented on the x-axis. The graph is a circle centred at (2, 0) with radius 2.

Step 1: Understand the Standard Form of Polar Equation

A polar equation of the form \(r^{2}=a^{2} \cos(2 \theta)\) represents a limaçon. The length \(a^{2}\) determines the size of the limaçon, and the form \(\cos\) or \(\sin\) determines the orientation. In this case, the polar equation \(r^{2}=4 \cos 2 \theta\) is a limaçon oriented on the x-axis.

Step 2: Transform into Cartesian Coordinates

To create a graph more easily, convert the polar equation into Cartesian coordinates. In Cartesian coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). Given the polar equation \(r^{2}=4 \cos 2 \theta\), it can be rewritten as \(x^{2}+ y^{2} = 4 x\).

Step 3: Sketch the Graph

The graph of the equation \(x^{2}+ y^{2} = 4 x\) is a circle centred at (2, 0) with radius 2. This circle is the limaçon described by the polar equation \(r^{2}=4 \cos 2 \theta\). Thus, the graph is a circle centred at (2, 0) with radius 2.