College Algebra, Chapter 2, Review Exercises, Section Review Exercises, Problem 12

a.) Use completing the square to determine whether the equation $\displaystyle 2x^2 + 2y^2 - 2x + 8y = \frac{1}{2}$ represents a circle or a point or has no graph.

$\begin{equation} \begin{aligned} 2x^2 + 2y^2 - 2x + 8y &= \frac{1}{2} && \text{Model}\\ \\ x^2 + y^2 - x + 4y &= \frac{1}{4} && \text{Divide both sides by } 2\\ \\ \left( x^2 - x + \underline{\phantom{xx }} \right) + \left( y^2 + 4y + \underline{\phantom{xx }} \right) &= \frac{1}{4} && \text{Group terms}\\ \\ \left( x^2 - x + \frac{1}{4} \right) + ( y + 4y + 4 ) &= \frac{1}{4} + \frac{1}{4} + 4 && \text{Complete the square: Add } \left( -\frac{1}{2} \right)^2 = \frac{1}{4} \text{ and } \left( \frac{4}{2} \right)^2 =4 \\ \\ \left( x - \frac{1}{2} \right)^2 + (y + 2)^2 &= \frac{9}{2} && \text{Perfect square, the equation represents a circle.} \end{aligned} \end{equation}$


b.) If the equation is part of a circle. Find its center and radius and sketch its graph.
The equation has center at $\displaystyle \left( \frac{1}{2}, -2 \right)$ and radius $\displaystyle r^2 = \frac{9}{2}, \quad r = \frac{3}{\sqrt{2}}$