Problem 29 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$r^{2}=9 \cos 2 \theta$$

Step-by-Step Solution

Answer

The curve of the given polar equation is symmetric about the origin and the graph will resemble an 'infinity' symbol or a 'Lemniscate'.

Step 1: Analyzing the Exercise

The given equation is of the form \(r^{2} = 9\cos(2\theta)\). It is clear that the equation is in polar coordinates and needs to be graphed. Before graphing the equation, it's important to verify if the curve is symmetrical.

Step 2: Checking for symmetry about the Polar Axis

Replace \(\theta\) by \(-\theta\) in the equation. If the equation remains the same then it is symmetric about the polar axis or x-axis. But here, when we replace \(\theta\) by \(-\theta\) the equation changes to \(r^{2} = 9\cos (-2\theta)\), which is not the same as the original. Therefore, the curve is not symmetric about the polar axis.

Step 3: Checking for symmetry about the Line θ=π/2

We replace \(\theta\) by \(\pi - \theta\). If the resulting equation matches the original one, then the curve is symmetric about the line \(\theta = \pi/2\) or y-axis. On replacing \(\theta\) by \(\pi - \theta\), the equation becomes \(r^{2} = 9\cos(2\pi - 2\theta)\), which does not match the original equation. Hence the curve is not symmetric about the line θ=π/2.

Step 4: Checking for symmetry about the Origin

Replace \(r\) with \(-r\) and see if we get the same equation. This checks symmetry with respect to origin. It's because, replacing \(r\) with \(-r\) and getting the same equation back means that all points (r, θ) and (-r, θ) (which are diametrically opposite each other) are on the graph. The equation remains \(r^{2} = 9\cos(2\theta)\), even when \(r\) is replaced with \(-r\). Therefore, the curve is symmetric with respect to the origin.

Step 5: Plotting the curve

Now that we've determined the symmetries prevalent in the graph, the next step is to plot the curve for \(r^{2} = 9\cos(2\theta)\). This is a Lemniscate, tha graph of which would be similar to an infinity symbol. Remember to make note that the equation is symmetric with respect to origin.