Problem 15 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

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

Step 1: Identifying the General Circle Equation

The general equation for a circle is \[ (x-h)^2 + (y-k)^2 = r^2 \]where \((h, k)\) are the coordinates of the center and \(r\) is the radius of the circle.

Step 2: Comparing with the Given Equation

Now, compare the given equation \(x^2 + y^2 = 9\) with the general form\[ (x-h)^2 + (y-k)^2 = r^2 \] to find the values of \(h\), \(k\), and \(r\). Here, \(h = 0\), \(k = 0\), and \(r^2 = 9\).

Step 3: Finding the Center

Since the given equation has no terms \((x-h)^2\) or \((y-k)^2\) involving non-zero \(h\) or \(k\), it implies that the center is at the origin, \((0, 0)\).

Step 4: Calculating the Radius

The equation gives us \(r^2 = 9\). Solving for \(r\), we find:\[ r = \sqrt{9} = 3 \]Thus, the radius of the circle is 3.

Step 5: Graphing the Circle

To graph the circle, plot the center at the origin \((0, 0)\) and draw a circle with a radius of 3 units in all directions. This creates a circle that crosses the points \((3, 0)\), \((-3, 0)\), \((0, 3)\), and \((0, -3)\).

Key Concepts

Understanding the Center of a Circle

The center of a circle is a crucial concept. It's simply the point inside the circle that is equidistant from every point on the circle's edge. Think of the center as the anchor of the circle.
In the equation \[ (x-h)^2 + (y-k)^2 = r^2 \],the center is represented by the coordinates \((h, k)\).This tells us exactly where the circle is located on a graph.

• If the equation looks like \(x^2 + y^2 = ... \),with no \(h\) or \(k\) values altering \(x\) or \(y\),it means the center is at the origin \((0, 0)\).
• Misplaced or altered values of \(h\) and \(k\)move the circle away from the origin in the coordinate plane.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle. It remains the same no matter which direction you measure. It is one of the most important parameters, as it defines the size of the circle.
The radius is represented by \(r\) in the circle's equation \((x-h)^2 + (y-k)^2 = r^2\).In the example equation \(x^2 + y^2 = 9\),we find \(r^2 = 9\),which leads to
\(r = \sqrt{9} = 3\).So, the circle's radius is 3 units long.

• A larger \(r\)means a larger circle.
• If \(r=0\),the circle reduces to a point, which is uncommon in these problems.

Graphing Circles

Graphing a circle involves plotting its center and using its radius to draw the circular boundary on the graph.
Start by locating the center of the circle on the coordinate plane. For our example, the center is at \((0, 0)\),which simplifies graphing significantly.
Then, measure out the radius from the center in every direction, basically as the circle's reach in the plane.
A circle with a radius of 3 reaches out to:

• (3, 0) and (-3, 0) on the x-axis
• (0, 3) and (0, -3) on the y-axis

Once these points are plotted, draw a smooth curve through them, completing the circle. This visual representation helps you understand the relationship between algebraic equations and geometric shapes. Remember:

• Treat the radius as the circle's leg span—it's how far it can stretch.
• Always check that your circle is proportionately round with respect to the axes.