Problem 23 - Intermediate Algebra

Question

The graph of each equation is a circle. Find the center and the radius and then graph the circle. $$x^{2}+y^{2}=9$$

Step-by-Step Solution

Answer

Center: (0, 0), Radius: 3.

Step 1: Identify the Equation Type

The given equation is in the form of \(x^2 + y^2 = r^2\), which represents a circle centered at the origin with a radius \(r\).

Step 2: Determine the Center

For the equation \(x^2 + y^2 = 9\), it can be rewritten as \((x-0)^2 + (y-0)^2 = 3^2\). Thus, the center of the circle is \((0, 0)\).

Step 3: Calculate the Radius

From the equation \(x^2 + y^2 = 9\), we see that \(r^2 = 9\). Solving for \(r\), we get \(r = \sqrt{9} = 3\). Thus, the radius of the circle is 3.

Step 4: Graph the Circle

Plot the center of the circle at \((0, 0)\) on a coordinate plane. From this center, use a compass to draw a circle with a radius of 3 units. This circle will intersect the points \((3, 0), (-3, 0), (0, 3), (0, -3)\) on the coordinate plane.

Key Concepts

center of a circle

The center of a circle in the Cartesian plane is a crucial point from which every point on the circle's boundary is equidistant. This is often denoted in a circle equation as \(x - h\)^2 + \(y - k\)^2 = r^2, where \(h, k\) represents the center. In simpler terms, the center is the balancing point of the circle.
The given exercise's equation \(x^2 + y^2 = 9\) is an example where the center is at the origin. By rewriting the equation as \( (x - 0)^2 + (y - 0)^2 = 3^2 \), we clearly identify that the center is at \(0, 0\).
Remember:

• The x-coordinate of the center is the value that x is "offset" by in the equation.
• The y-coordinate is similarly the offset value in the equation for y.

For standard equations like \(x^2 + y^2 = r^2\), the center is always \(0, 0\). That makes it easy to visualize and plot in the graph.

radius of a circle

The radius of a circle is the distance from its center to any point on its boundary. It's a constant measurement throughout a circle.
In the equation form \(x^2 + y^2 = r^2\), the value of \(r^2\) directly gives the squared radius of the circle. To find the actual radius, we take the square root of this value.
For the exercise, the equation \(x^2 + y^2 = 9\) gives us \(r^2 = 9\). Calculating the square root, we find that the radius \(r\) is 3.
Keep in mind:

• The diameter of the circle is twice the radius. So, if the radius is 3, the diameter would be 6.
• The radius remains the same for any point on the circle's boundary.

The understanding of the radius not only helps in solving equations but also plays a crucial role in graphing circles.

graphing circles

Graphing circles involves plotting a perfect closed curve where all points are equidistant from the center. Here's a simple step-by-step:
Start by plotting the center of the circle on a coordinate plane. For our exercise, the center is at \(0, 0\).
Next, use the radius to mark key points in each direction from the center:

• Move right by the radius (3 units) to plot point \(3, 0\).
• Move left by the radius (3 units) to plot point \(-3, 0\).
• Move up by the radius (3 units) to plot point \(0, 3\).
• Move down by the radius (3 units) to plot point \(0, -3\).

These points will give you the boundary references of your circle. You can use a compass, or draw free-hand, to connect these points smoothly. This ensures a perfect circular representation is made on your graph.