Problem 28 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \((5,-4) ;\) radius 6

Step-by-Step Solution

Answer

The equation of the circle is \((x - 5)^2 + (y + 4)^2 = 36\).

Step 1: Understanding Circle's Equation

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Substituting the Center

Given the center of the circle \((h, k) = (5, -4)\), substitute \(h = 5\) and \(k = -4\) into the standard form equation, giving \((x - 5)^2 + (y + 4)^2 = r^2\).

Step 3: Substituting the Radius

Given the radius \(r = 6\), substitute \(r = 6\) into the equation, resulting in \((x - 5)^2 + (y + 4)^2 = 6^2\).

Step 4: Final Equation

Simplify \(6^2\) to obtain \((x - 5)^2 + (y + 4)^2 = 36\). This is the standard form of the circle's equation.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is a fundamental concept in geometry. It is expressed as \((x - h)^2 + (y - k)^2 = r^2\). This equation is so useful because it clearly shows the circle's center and radius at a glance. When you see an equation in this format, you immediately know key details about the circle.

• \((h, k)\) represents the center of the circle, giving you the precise location of its midpoint.
• \(r\) stands for the radius, which is the distance from the center to any point on the circle.

The equation might look a bit complex at first, but it breaks down the circle's position and size very clearly with just those elements. With the standard form, identifying the circle's properties becomes straightforward.

Center of a Circle

Understanding the center of a circle is crucial. The center is designated by the coordinates \((h, k)\) in the standard form of the circle. It marks the exact middle point from which the circle spreads out.
To find the center, just look at the values inside the parentheses in the equation:

• In the equation \((x - h)^2 + (y - k)^2 = r^2\), the center is \((h, k)\).
• For example, if you have an equation like \((x - 5)^2 + (y + 4)^2 = 36\), the center is \((5, -4)\).

The center is integral because it defines where the circle is located on a graph. No matter where the radius reaches, everything revolves around this central point.

Radius of a Circle

The radius is a vital part of a circle's definition and gives the circle its size. In the standard form equation, the term \(r^2\) represents the square of the radius.
Understanding the radius is simple:

• The radius \(r\) is the distance from the center to any point on the circle.
• It's the length of the line segment from \((h, k)\) to any point \((x, y)\) on the circle's edge.
• If your equation is \((x - 5)^2 + (y + 4)^2 = 36\), the radius \(r\) is 6 because \(r^2 = 36\).

The radius determines just how big the circle is. When you substitute the radius into the standard form equation, you can immediately calculate how far the circle reaches from its center.