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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q. 9 (page 840)

Find the polar equation of a circle with center at $(0, 4)$ that passes through the pole. What is the rectangular equation?

Short Answer

Expert verified

The polar equation of the circle is $x^{2} + y^{2} - 8 y = 0$ and rectangular equation is $r - 8 \sin 胃 = 0 .$

Step by step solution

01

Step 1. Given Information

Centre of the circle is $(0, 4)$ that passes through the pole. This means that radius is 4.

We need to find the equation of circle.

02

Step 2. Finding Equation of circle

${(x - h)}^{2} + {(y - k)}^{2} = r^{2} {(x - 0)}^{2} + {(y - 4)}^{2} = 4^{2} x^{2} + y^{2} + 16 - 8 y = 16 x^{2} + y^{2} - 8 y = 0$

03

Step 3. Rectangular Equation

Put $x = r \cos 胃, y = r \sin 胃$

$x^{2} + y^{2} - 8 y = 0 r^{2} - 8 r \sin 胃 = 0 r (r - 8 \sin 胃) = 0 r - 8 \sin 胃 = 0$

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