Problem 74 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. Complete the square and write the equation in standard form: \(x^{2}+6 x+y^{2}=0 .\) Then give the center and radius of the circle, and graph the equation.

Step-by-Step Solution

Answer

The standard form of the given equation is \((x+3)^{2} + y^{2} = 9\). The center of the circle is \((-3,0)\) and the radius is 3.

Step 1: Complete the square for the x-terms

First, group the \(x\) terms together, leaving the \(y^{2}\) term on the other side of the equation: \(x^{2}+6x = -y^{2}\). Then, add the square of half the coefficient of \(x\) to both sides of the equation. In this case, that will be \((6/2)^{2} = 9\), giving the equation: \(x^{2}+6x+9 = -y^{2}+9\)

Step 2: Write the equation in standard form and identify the center and radius

Rewrite the left side of the equation as a binomial squared and simplify the right side: \((x+3)^{2} = 9 - y^{2}\). Then, add \(y^{2}\) to both sides of the equation to make the \(y\) term positive, resulting in \((x+3)^{2} + y^{2} = 9\). This is the standard form of a circle. This means the center of the circle is at \((-3,0)\) and the radius is \(r = \sqrt{9} = 3\).\n

Step 3: Graph the circle

Mark the center of the circle at \((-3,0)\). Then from the center, draw a circle with radius 3. Your circle should pass through the points \((0,0), (-3,3), (-6,0)\), and \((-3,-3)\).

Key Concepts

Circle Equation

When dealing with circles in algebra, you'll often encounter their equations. The general equation for a circle is \[ (x - h)^2 + (y - k)^2 = r^2 \]
where

• \( h \) is the x-coordinate of the center of the circle,
• \( k \) is the y-coordinate of the center of the circle,
• \( r \) is the radius of the circle.

To solve for this form, you might need to "complete the square" so the equation can resemble the circle's standard form.
For example, starting with \( x^2 + 6x + y^2 = 0 \), you need to group and rearrange the \( x \) terms into a square.
By completing the square, you change the expression to \( (x+3)^2 + y^2 = 9 \). This process is essential because it allows you to clearly identify the circle's characteristics and graph it easily.

Center and Radius of a Circle

Understanding the center and radius of a circle is crucial when graphing circles and interpreting their equations.
Once you have rewritten the equation in standard form, like in \( (x+3)^2 + y^2 = 9 \), the values within the parentheses \( (x - h)^2 + (y - k)^2 \) tell you the circle's center: \( (h, k) \).

• Here, the center is \( (-3, 0) \).
• The number on the other side of the equation \( r^2 \) gives you the radius's square. For our example, \( r^2 = 9 \), thus the radius \( r \) equals \( \sqrt{9} = 3 \).

Being able to quickly identify these key features enables you to understand and manipulate circular shapes mathematically.

Graphing Circles

Graphing circles becomes straightforward once you've identified the center and radius from the equation.
First, locate the center on a coordinate plane — for the equation \( (x+3)^2 + y^2 = 9 \), plot the point \( (-3, 0) \).
Next, you can draw the circle by measuring the radius out from the center in all directions.
Since the radius is \( 3 \), place points at a distance of \( 3 \) units horizontally and vertically from the center point. These points might include \( (0, 0) \), \( (-3, 3) \), \( (-6, 0) \), and \( (-3, -3) \).
Finally, connect the points smoothly to form a circle, ensuring the circle intersects these points accurately.
This graphical representation visually demonstrates the relationships expressed in the circle's equation and helps solidify your understanding of circular geometry.