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Chapter 6: Problem 66

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=6 \sec \theta$$

Short Answer

Expert verified
After converting the polar equation \( r=6 \sec \theta \) to rectangular form \( x^4 - 36x^2 = 0 \), we plotted the resultant rectangular equation to create a vertical line graph at \( x = -6, 0, 6 \).

Step by step solution

01

Convert Polar Equation to Rectangular Equation

Given a polar equation \( r = 6 \sec \theta \), to convert this to a rectangular equation, recall that \( \sec \theta = 1/ \cos \theta \) and \( r = \sqrt{x^2 + y^2} \). So, substitute \( \sec \theta \) and \( r \) to get the rectangular equation as \( \sqrt{x^2 + y^2} = \frac{6}{\cos \theta} \) or \( x^2 + y^2 = \frac{36}{\cos^2 \theta} \). Now replace \( \cos^2 \theta \) with \( \frac{x^2}{r^2} \) to get \( x^2 + y^2 = 36 \cdot \frac{r^2}{x^2} \) simplifying we get \( x^4 = 36x^2 \) or \( x^4 - 36x^2 = 0 \). So finally, we've converted the polar equation to the rectangular equation \( x^4 - 36x^2 = 0 \)
02

Plotting of Rectangular Equation

To plot the equation \( x^4 - 36x^2 = 0 \), set it equal to zero to find the x coordinates where the graph crosses the x-axis. Solving for x, the equation decomposes into \( (x^2-6^2)(x^2) = 0 \) and the roots of this equation are \( x=-6, 0, 6 \). The general form of the equation is a quartic polynomial, which typically produces a W-shaped curve. Since the equation includes no \( y \) term, the function is entirely vertical for each value of \( x \). So plot vertical lines at \( x=-6, 0, 6 \).
03

Find zero points

By setting \( y=0 \) in the equation \( x^4 - 36x^2 = 0 \), it can be seen that the solution \( x^4 - 36x^2 = 0 \) holds, so the graph intersects the x-axis.

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