Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.9 - The Coordinate Plane; Graphs of Equations; Circles - 1.9 Exercises - Page 104: 101

Answer

The given equation represents a circle witih: center at $(\frac{1}{4}, -\frac{1}{4})$ radius = $\frac{1}{2}$

Work Step by Step

RECALL: The standard form of the equation of a circle whose center is at (h, k) and radius $r$ is: $(x-h)^2+(y-k)^2=r^2$ If the given equation can be written in the form given above, then it must represent a circle. Rewrite the given equation by completing the square to have: $\\(x^2-\frac{1}{2}x)+(y^2+\frac{1}{2}y)=\frac{1}{8} \\(x^2-\frac{1}{2}x+\frac{1}{16})+(y^2+\frac{1}{2}y+\frac{1}{16})=\frac{1}{8} + \frac{1}{16}+\frac{1}{16} \\(x-\frac{1}{4})+(y+\frac{1}{4})^2=\frac{2}{16}+\frac{1}{16}+\frac{1}{16} \\(x-\frac{1}{4})+(y+\frac{1}{4})^2=\frac{1}{4}$ Thus, the given equation represents a circle with center at $(\frac{1}{4}, -\frac{1}{4})$ and a radius of $\frac{1}{2}$ units.
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