Problem 24 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ x y=4 $$

Step-by-Step Solution

Answer

The polar form of given rectangular equation is \(r^2 = \frac{8}{\sin(2\theta)}\) if \(\sin(2\theta) ≠ 0\). The graph is a lemniscate, which forms loops in the positive and negative regions.

Step 1: Converting rectangular equation to polar form

Replace \(x\) and \(y\) in the given equation \(xy = 4\) by corresponding polar forms. So, \(r\cos(\theta) \cdot r\sin(\theta) = 4\). This leads to \(r^2 \sin(\theta) \cos(\theta) = 4\).

Step 2: Simplifying the polar equation

The double-angle identity for \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\) can be used to simplify the equation. With this identity, the equation then becomes \(r^2 \sin(2\theta) = 8\). This can be further simplified to \(r^2 = \frac{8}{\sin(2\theta)}\) if \(\sin(2\theta) ≠ 0\).

Step 3: Sketching the Polar Graph

This graph represents a lemniscate when \(r ≠ 0\) and \(\sin(2\theta) ≠ 0\). The graph passes the origin at \(\theta = 45°\) and \(\theta = 135°\). The loop in the positive region is contained within \(0 < \theta < 90°\), and the loop in the negative region is contained within \(90° < \theta < 180°\).