Problem 25 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((6,0) ;\) radius \(=4\)

Step-by-Step Solution

Answer

The equation of the circle with the given center and radius is \[\boxed{(x-6)^2 + y^2 = 16}\].

Step 1: Identify the center and radius

The given center is \((6,0)\) which means \(h=6\) and \(k=0\). The given radius is \(4\).

Step 2: Substitute the known values into the equation of a circle

Now, substitute the values \(h=6, k=0\), and \(r=4\) into the circle equation: \((x-h)^2 + (y-k)^2 = r^2\)

Step 3: Simplify the equation and obtain the final result

Substitute the values into the equation: \((x-6)^2 + (y-0)^2 = 4^2\) Simplify and we have: \((x-6)^2 + y^2 = 16\) Thus, the equation of the circle with the given center and radius is: \[\boxed{(x-6)^2 + y^2 = 16}\]

Key Concepts

Center and Radius of a Circle

In circle geometry, the center is the point right in the middle of the circle. It is usually represented by the coordinates \((h, k)\). This tells you exactly where the circle is located on a coordinate plane. For this exercise, the given center is \((6, 0)\), meaning the circle is 6 units to the right of the origin and sits on the x-axis.

• Center: \((h, k) = (6, 0)\)
• Radius: The radius is the distance from the center to any point on the circle. It measures how big the circle is. Here, the radius is 4 units long.

Understanding the center and radius is key because knowing these allows us to write the equation of the circle.

Standard Form of a Circle Equation

The equation of a circle in standard form is a mathematical representation of the circle using its center and radius. The standard equation is given by:\[(x - h)^2 + (y - k)^2 = r^2\]Here:

• \((h, k)\) represents the coordinates of the circle's center.
• The term \(r\) is the radius of the circle.

This equation is very useful because it provides a way to generalize and write the equation for any circle as long as you have the center and the radius. It shows the relationship between the x and y coordinates of any point on the circle to the center and the radius.

Substituting Values into Circle Equation

Once you have the center and radius, it's time to substitute them into the circle's standard equation to find the specific equation for your circle.
This is how you translate the geometric details into a precise algebraic expression.

• Given: Center \((6, 0)\), radius = 4
• Substitute into the template: \((x-6)^2 + (y-0)^2 = 4^2\)
• Simplify the right side: \(4^2 = 16\)

This substitution process transforms the standard form into a definite formula for your circle: \[(x-6)^2 + y^2 = 16\]The simplification provides the circle's specific equation, which identifies all points (x, y) that lie on the circle.