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Chapter 6: Problem 72

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=8 \cos \theta+2 \sin \theta$$

Short Answer

Expert verified
The rectangular form of the equation is \((x - 4)^2 + (y - 1)^2 = 17\). The graph is a circle centered at (4,1) with radius \(\sqrt{17}\).

Step by step solution

01

Replace \(r\) in terms of \(x\) and \(y\)

We first need to rewrite \(r\) using Pythagorean theorem as \(r = \sqrt{x^2 + y^2}\). This gives us the equation \(\sqrt{x^2+y^2}=8 \cos \theta+2 \sin \theta\).
02

Replace \(\cos \theta\) and \(\sin \theta\)

Next, replace \(\cos \theta = x/r\) and \(\sin \theta = y/r\) in the equation. Using the replacements, we have \(\sqrt{x^2+y^2} = 8(x / \sqrt{x^2 + y^2}) + 2(y / \sqrt{x^2 + y^2})\).
03

Simplify the equation

Now, cross multiply and simplify to get rid of the square root in the equation. This yields the rectangular equation \(x^2 - 8x + y^2 - 2y = 0\).
04

Complete the square

To express the equation in standard form, complete the square for both \(x\) and \(y\). As such, the equation becomes \((x - 4)^2 + (y - 1)^2 = 17\).
05

Graph the rectangular equation

Finally, plot the equation \((x - 4)^2 + (y - 1)^2 = 17\) which forms a circle centered at (4,1) with radius \(\sqrt{17}\) in the rectangular coordinate system.

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