Problem 12 - Beginning and Intermediate Algebra

Question

Identify the center and radius of each circle and graph. $$x^{2}+y^{2}=16$$

Step-by-Step Solution

Answer

The center of the circle is at the point (0, 0) and its radius is 4 units. To graph the circle, plot the center at (0, 0) and draw a circle with a radius of 4 units, passing through the points (4, 0), (0, 4), (-4, 0), and (0, -4).

Step 1: Convert the given equation into the standard form of a circle

The given equation is already in the standard form of a circle: \[(x-0)^2 + (y-0)^2 = 4^2\] Notice that \(a = 0\) and \(b = 0\), so our equation can be written as: \[(x^2) + (y^2) = 16\]

Step 2: Identify the center of the circle

From the standard form of the circle equation, we can identify the center of the circle as (a, b). In our case, \(a = 0\) and \(b = 0\), so the center of the circle is at the point (0, 0).

Step 3: Identify the radius of the circle

To find the radius of the circle, we need to look at the right side of the equation. In our case, it is \(4^2 = 16\). This tells us that the radius of the circle is \(r = 4\).

Step 4: Graph the circle

To graph the circle, follow these steps: 1. Plot the center of the circle at the point (0, 0). 2. From the center, draw a circle with a radius of 4 units. The circle should pass through the points (4, 0), (0, 4), (-4, 0), and (0, -4).

Key Concepts

Standard Form of a Circle

In geometry, the standard form of a circle provides an easy way to define and understand the position and size of a circle on a 2D plane. The standard form equation is written as: \[(x-a)^2 + (y-b)^2 = r^2\]Here,

• \( (a, b) \) are the coordinates of the center of the circle.
• \( r \) is the radius of the circle.

This form allows for easy identification of both the center and the radius, which are crucial for graphing and understanding the circle's properties. In our example, the equation \( x^2 + y^2 = 16 \) can be rewritten to show \[(x-0)^2 + (y-0)^2 = 4^2\] indicating that the circle has a center at the origin \( (0, 0) \) and a radius of \( 4 \).

Center of a Circle

The center of a circle is a point that is equidistant from all points on the circle's edge. It is denoted as \((a, b)\) in the circle's standard form equation. This point is fundamental in determining the circle's location on a graph.
In our specific case, we see that the equation \(x^2 + y^2 = 16\) simplifies to \[(x-0)^2 + (y-0)^2 = 4^2\]indicating the circle's center is at the origin \((0, 0)\).
The knowledge of a circle's center helps us plot the circle accurately, ensuring it is placed correctly on the coordinate plane.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. In the standard form equation, it appears as \(r\) in \((x-a)^2 + (y-b)^2 = r^2\). Understanding the radius helps in visualizing the circle's size.

• In our example \(x^2 + y^2 = 16\), when rewritten, it becomes\[(x-0)^2 + (y-0)^2 = 4^2\]
• This tells us that the radius \(r\) is \(4\).

Knowing the radius is essential for drawing the circle accurately, ensuring its correct size. This simple value also helps calculate the circle's area and circumference if needed.

Graphing Circles

Graphing a circle involves translating the mathematical equation into a visual representation on a coordinate plane. This process allows us to clearly see the circle's size and position.
Here are simple steps to graph a circle:

• First, identify and plot the center point \((a, b)\).
• Using the radius \(r\), measure and mark points at distance \(r\) from the center point in multiple directions.
• Connect these points smoothly to form the circle.

In our example, with the equation \(x^2 + y^2 = 16\),

• Plot the center at \((0, 0)\).
• Next, use a radius of \(4\) to mark points at \((4, 0), (0, 4), (-4, 0), \text{and} (0, -4)\).
• Draw a smooth curve through these points to complete the circle.

This approach allows for accurate and clear graphing of the circle on any coordinate plane.