Problem 29 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$x^{2}+y^{2}+6 y=0$$

Step-by-Step Solution

Answer

The center is (0, -3) and the radius is 3.

Step 1: Rearrange the Equation

The given equation of the circle is \( x^2 + y^2 + 6y = 0 \). First, isolate the terms involving \( y \) and write the equation as \( x^2 + (y^2 + 6y) = 0 \).

Step 2: Complete the Square for y

To complete the square for the term involving \( y \), take half of the coefficient of \( y \), which is 6, and square it. Half of 6 is 3, and 3 squared is 9. Add and subtract 9 inside the equation: \( x^2 + (y^2 + 6y + 9 - 9) = 0 \).

Step 3: Simplify the Equation

Now simplify the equation from the previous step: \( x^2 + ((y + 3)^2 - 9) = 0 \). This can be rewritten as \( x^2 + (y + 3)^2 = 9 \).

Step 4: Identify the Center and Radius

The equation \( x^2 + (y + 3)^2 = 9 \) is of the form \((x-h)^2 + (y-k)^2 = r^2\). Here, \( h = 0 \), \( k = -3 \), and \( r^2 = 9 \), hence \( r = 3 \). Therefore, the center of the circle is \((0, -3)\) and the radius is 3.

Step 5: Graph the Circle

To graph the circle, plot the center at the point \((0, -3)\) on the coordinate plane. Using the radius of 3, draw a circle around this center, ensuring that all points on the circle are 3 units away from the center.

Key Concepts

Understanding the Center of a Circle

When dealing with the equation of a circle, identifying the center is crucial in helping you understand where the circle is located on the coordinate plane. The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center.

• **\(h\)**: This represents the x-coordinate of the circle's center. In our example equation, since there is no \( (x - h) \) term, \( h = 0 \).
• **\(k\)**: This is the y-coordinate of the center. The term \((y + 3)\) suggests that \( k = -3 \).

Therefore, the center of the circle in our exercise is \((0, -3)\). This tells us exactly where the middle of the circle lies in our coordinate system. Understanding these coordinates helps in visualizing and accurately plotting the circle on a graph.

Determining the Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle. After finding the center, the next step is to determine the radius using the circle's equation.
In the equation \((x - h)^2 + (y - k)^2 = r^2\), the term \(r^2\) represents the square of the radius. To find \(r\), you take the square root of \(r^2\).In our equation, we have \((x - 0)^2 + (y + 3)^2 = 9\). Here, \(r^2 = 9\), which means:

• Take the square root of 9, giving \(r = 3\).

The radius is thus 3 units. Understanding the radius is important because it dictates how large the circle will be when you draw it on the graph. You can imagine the circle expanding outward from its center point until every outer point is exactly 3 units away.

Completing the Square to Simplify Circle Equations

Completing the square is a mathematical technique used to simplify quadratic expressions and make equations easier to work with.
For circle equations, it helps transform expressions into the standard form \((x-h)^2 + (y-k)^2 = r^2\).Here's how to complete the square:

• Look at the term \(y^2 + 6y\). Take half of the coefficient of \(y\) which is 6, so half of it is 3.
• Square this number (\(3^2 = 9\)) and add and subtract it within the expression to keep the equation balanced: \(x^2 + (y^2 + 6y + 9 - 9) = 0\).
• Rewrite this as \( (y + 3)^2 - 9 \) to simplify: \(x^2 + (y + 3)^2 = 9\).

By completing the square, you can turn complex expressions into ones that reveal essential features of the circle. It's an invaluable tool for recognizing a circle's center and its radius, as shown in our example.