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Chapter 2: Problem 64

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}+3 x+5 y+\frac{9}{4}=0$$

Short Answer

Expert verified
The standard form of the equation is \((x+ \frac{3}{2})^{2} + (y + \frac{5}{2})^{2} = 1\). The center and radius of the circle are \((- \frac{3}{2}, - \frac{5}{2})\) and 1 respectively.

Step by step solution

01

Rearranging the equation

The first step is to rearrange the given equation to make the completion of the square easier. This means gathering the \(x\) and \(y\) terms together, and moving the constant to the other side of the equality. The equation becomes: \((x^{2}+3x)+(y^{2}+5y) = -\frac{9}{4}\)
02

Completing the square

Next, we complete the square for each variable. Starting with \(x\), we take half of the coefficient of \(x\), square it, and add to both sides of the equation. Doing the same for \(y\), The equation becomes: \((x+ \frac{3}{2})^{2} - \frac{9}{4} + (y + \frac{5}{2})^{2} - \frac{25}{4} = -\frac{9}{4}\). Moving all constants to the right side of the equation, we get \((x+ \frac{3}{2})^{2} + (y + \frac{5}{2})^{2} = 1\)
03

Finding the center and radius

The center of the circle is \((- \frac{3}{2}, - \frac{5}{2})\). The radius is given by the square root of the constant on the right side of the equation, which in this case is \(r=1\)
04

Graphing the equation

To graph this equation, mark the center of the circle at point \((- \frac{3}{2}, - \frac{5}{2})\). Then, draw a circle with radius 1, making sure all points on the circle are 1 unit away from the center. Make sure to include the x-axis and y-axis, and choose appropriate scales on both axes.

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Most popular questions from this chapter

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