/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Convert each polar equation to a... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 69

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=12 \cos \theta$$

Short Answer

Expert verified
The rectangular equation is \( (x-6)^2 + y^2 = 36 \). It represents a circle centered at (6,0) with radius 6 units.

Step by step solution

01

Conversion to rectangular form

In polar coordinates, \( r = x\cos\theta + y\sin\theta \). Multiplying both sides of the given equation \( r =12\cos\theta \) by r we have \( r^2 = 12r\cos\theta \). As we know that \( r^2 = x^2 + y^2 \) and \( r\cos\theta = x \) in rectangular coordinates, we replace these in our polar equation. That gives us the rectangular equation \( x^2 + y^2 = 12x \) which can be simplified to \( x^2 - 12x + y^2 = 0 \)
02

Completing square to reformat equation

For the ease of graphing, the equation can be rearranged into the form of a circle equation \( (x-h)^2 + (y-k)^2 = r^2 \) where (h,k) is the center of the circle and r is the radius. For that, complete the square for x terms. Taking \( y^2 \) to the other side result in \( x^2 - 12x = -y^2 \). Adding 36 to both sides complete the squares which give us \( (x-6)^2 = -y^2 + 36 \)
03

Getting the equation of the circle

Finally, in order to have the equation resemble the circle's standard form, we add \( y^2 \) on both sides which gives us \( (x-6)^2 + y^2 = 36 \). This is the equation of circle centered at (6,0) with radius 6 units.
04

Graphing the equation

Plot the center of the circle at (6,0). Draw a circle with radius 6 units. This is the graph of given polar equation in rectangular coordinates.

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