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Chapter 2: Problem 61

Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. $$x^{2}+y^{2}-x+2 y+1=0$$

Short Answer

Expert verified
The center of the circle is at point (-0.5, -1) and the radius is approximately 1.12 units.

Step by step solution

01

Step 1. Arrange and complete square for x terms

Arrange the terms containing \(x\) together, which are \(x^2 - x\). In order to complete the square, add and subtract \((-\frac{1}{2})^2 = 0.25\) to the equation.\nSo, the new equation becomes \(x^2 - x + 0.25 + y^2 +2y +1 - 0.25 =0\).
02

Step 2. Arrange and complete square for y terms

Next, arrange the terms containing \(y\) together, which are \(y^2 + 2y\). To complete the square, add and subtract \((\frac{2}{2})^2 = 1\) to the equation.\nThe resulting equation becomes \(x^2 - x + 0.25 + y^2 +2y +1 - 0.25 + 1 - 1 =0\).
03

Step 3. Write the equation in standard form

The equation now can be written as \([x - (-0.5)]^2 + [y - (-1)]^2 = 0.5^2 + 1^2\). Which simplifies to \([x + 0.5]^2 + [y +1]^2 = 1.25\). This is the standard form.
04

Step 4. Identify the center and the radius

From the standard form, the center of the circle is at point \((-0.5, -1)\) and the radius of the circle is \(\sqrt{1.25} \approx 1.12\) units.
05

Step 5. Graph the equation

Plot the center point on a graph, use the radius to draw a circle around the center. Be sure to label the center and also the distance of the radius. This now represents the circle for the given equation.

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Most popular questions from this chapter

Exercises \(103-105\) will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center \((1,-1)\) and radius 1.

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