Problem 28 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((1,0) ;\) radius \(=3 \sqrt{2}\)

Step-by-Step Solution

Answer

The equation of the circle with center \((1,0)\) and radius \(3\sqrt{2}\) is \((x-1)^2 + y^2 = 18\).

Step 1: Write down the standard equation of a circle

The standard equation of a circle with center \((h,k)\) and radius \(r\) is: \[(x-h)^2 + (y-k)^2 = r^2\]

Step 2: Substitute the given values

Now we plug in the center \((1,0)\) and radius \(3\sqrt{2}\) into the equation: \[(x-1)^2 + (y-0)^2 = (3\sqrt{2})^2\]

Step 3: Simplify the equation

Simplify the equation, taking note that the term \((3\sqrt{2})^2\) evaluates to \(18\): \[(x-1)^2 + y^2 = 18\] The equation of the circle with center \((1,0)\) and radius \(3\sqrt{2}\) is \((x-1)^2 + y^2 = 18\).

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is a fundamental concept in algebra and geometry. This equation provides a convenient way to mathematically describe a circle on a coordinate plane. The general form is \((x-h)^2 + (y-k)^2 = r^2\). Here, \((h, k)\) represents the coordinates of the circle's center, and \(r\) is the radius of the circle.

• The left-hand side of the equation involves \((x-h)^2\) and \((y-k)^2\), indicating that these terms account for the distance between any point \((x, y)\) on the circle and the center \((h, k)\).
• The right-hand side, \(r^2\), is simply the square of the circle’s radius. It determines the size of the circle.

Understanding this equation is crucial because it enables us to graph circles effortlessly by identifying their centers and radii, as well as solving problems involving geometric shapes.

Center of a Circle

The center of a circle is the fixed point from which all the points on the circumference are equidistant. In the context of the standard equation, the center is denoted by \((h, k)\). These two components, \(h\) and \(k\), are the x-coordinate and y-coordinate of the center, respectively.

• In our example with center \((1,0)\), the center of the circle is located at \(x = 1\) and \(y = 0\).
• This defines the position of the circle on the Cartesian plane. It acts as the anchor or pivot point around which the circle is perfectly symmetrical.

Knowing the center helps in determining the location and orientation of the circle on a grid. It is a critical component for plotting and for applications involving symmetry and radial balance.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its boundary. It determines the size of the circle and appears as \(r\) in the standard equation \((x-h)^2 + (y-k)^2 = r^2\). When squared, it completes the equation’s right side.

• In this particular exercise, the radius is \(3\sqrt{2}\), which equals approximately \(4.24\) when calculated. However, it’s crucial to work with precise values like \(3\sqrt{2}\) when performing algebraic operations.
• To compute \(r^2\), we use \((3\sqrt{2})^2 = 18\). This represents the radius squared, forming the equation part that balances with the left side denominating the squared terms for \(x\) and \(y\).

Proper understanding of the radius is vital in determining the circle's size and in calculations involving its area and perimeter (circumference). It serves as an important metric in designing and analyzing circular shapes.