Problem 29 - Algebra 2

Question

Use the center and the radius to graph each circle. $$ (x-6)^{2}+y^{2}=64 $$

Step-by-Step Solution

Answer

The graphed circle will have its center at point (6, 0) and a radius of 8 units, intersecting the coordinate plane at points (14, 0), (-2, 0), (6, 8), and (6, -8).

Step 1: Identify the Center and Radius

From the equation \((x-6)^{2} + y^{2} = 64\), we can identify the center as (h, k) = (6, 0) and the radius r as \(\sqrt{64}\) = 8.

Step 2: Sketch the Center

Plot the center of the circle at point (6, 0) on the coordinate plane.

Step 3: Draw the Circle

Using the center as the starting point, draw a circle with a radius of 8 units. This circle will intersect the coordinate plane at points (6+8, 0), (6-8, 0), (6, 0+8), and (6, 0-8).

Key Concepts

Center of a Circle

The center of a circle is a pivotal point from which every point on the circle is equidistant. In mathematical terms, the center of a circle is represented as a coordinate point \((h, k)\) in the equation of a circle, which is in the form \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) directly tells us where the circle's center is located on the coordinate plane.
For example, in the equation \((x-6)^2 + y^2 = 64\), we see that \((h, k) = (6, 0)\). This means the center of the circle is positioned at point (6, 0) on the coordinate plane.

• The x-coordinate : 'h' shifts the center along the x-axis.
• The y-coordinate : 'k' shifts the center along the y-axis.

Understanding the center helps us know where to start drawing the circle when graphing.

Radius of a Circle

The radius of a circle is a line segment that joins the center of the circle to any point on its perimeter. It's a crucial measure that defines the size of the circle.
In an equation like \((x-6)^2 + y^2 = 64\), the radius squared is represented by the number on the right side of the equation. So, in this case, 64 is the square of the radius. To find the actual radius \(r\), we calculate the square root:
\[r = \sqrt{64} = 8\]
This tells us that from the center (6, 0), every point on the edge of the circle is 8 units away.

• The radius determines how large the circle will be.
• It's an essential parameter for plotting the circle accurately.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional real number graph defined by a horizontal x-axis and a vertical y-axis. It's an essential tool for graphing circles and other geometric shapes. When plotting a circle, understanding how this plane works is crucial.
The center coordinates \((h, k)\) and radius \(r\) will guide you in accurately graphing the circle on this plane.

• Point (0,0) is the origin where both axes intersect.
• Coordinates are marked as (x, y) on this plane.

So, when we place the center of our circle at (6, 0), we move 6 units right on the x-axis and intersect the y-axis at 0.

Equation of a Circle

The equation of a circle in the standard form is \((x-h)^2 + (y-k)^2 = r^2\). This equation representation is key to understanding circle characteristics like center and radius.
The parts to recognize are:

• \((x-h)^2\) adjusts the circle's horizontal placement.
• \((y-k)^2\) adjusts the vertical placement via the y-coordinate of the center.
• \(r^2\) on the right side represents the radius squared.

In applying this to our example, \((x-6)^2 + y^2 = 64\), the elements (x-6) and y highlight the circle shifted 6 units right along the x-axis with a center on the y-axis. The 64 on the right side indicates that the radius is 8 units. This form directly guides us in plotting and understanding circle properties on the coordinate plane.