Problem 71 - College Algebra with Modeling and Visualization

Question

Find the standard equation of a circle that satisfies the conditions. Radius \(8,\) center \((3,-5)\)

Step-by-Step Solution

Answer

The standard equation is \\[ (x - 3)^2 + (y + 5)^2 = 64 \\].

Step 1: Understand the Standard Equation of a Circle

The standard form of the equation of a circle with center \(h, k\) and radius \r\ is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \]. In this problem, the center \((h, k)\) is \(3, -5\) and the radius \(r\) is \8\.

Step 2: Substitute the Center Coordinates into the Equation

Substitute the center coordinates \h = 3\ and \k = -5\ into the standard equation. This gives us: \[ (x - 3)^2 + (y + 5)^2 = r^2 \].

Step 3: Substitute the Radius into the Equation

Now, substitute the radius \(r = 8\) into the equation from Step 2. \(r^2 = 8^2 = 64\). This updates the equation to: \[ (x - 3)^2 + (y + 5)^2 = 64 \].

Step 4: Write the Final Equation

Combine the results from Steps 2 and 3 to write the final standard equation of the circle. The equation is: \[ (x - 3)^2 + (y + 5)^2 = 64 \].

Key Concepts

Standard Equation of a Circle

Understanding the standard equation of a circle is key to solving many geometric problems. The equation expresses the relationship between any point \((x, y)\) on the circle and its center \((h, k)\) with radius \(r\). The standard form is written as:
\[(x - h)^2 + (y - k)^2 = r^2\]You can think of this equation as saying "distance from the center." It tells us how far any point \((x, y)\) is from the circle's center. The left side of the equation represents the squared distance between \((x, y)\) and \((h, k)\). The right side, \(r^2\), is the square of the circle's radius.
This form's simplicity originates from the power of squaring differences. It ensures that all points \((x, y)\) satisfying the equation are exactly \(r\) units away from \((h, k)\), forming a perfect circle.

Radius and Center

The radius and center form the backbone of a circle's standard equation. In our context, the center of a circle is simply the fixed point \((h, k)\) from which every point on the circle maintains a constant distance, called the radius \(r\). For any circle:

• The center is expressed as \((h, k)\).
• The radius is a positive number, \(r\), indicating the circle's size.

For instance, a circle with a center at \((3, -5)\) and radius \(8\) would indicate:

• The circle is centered at \(3\) units along the x-axis and \(-5\) units along the y-axis.
• Every point on the edge of this circle is \(8\) units away from this center.

These two parameters allow us to accurately pinpoint a circle's position and size in a coordinate plane.

Substitution of Values

Substituting the values into the standard equation of a circle is a straightforward but critical step. This involves placing the given values of the radius and center coordinates into the equation. This substitution allows us to concretely identify the specific circle being described.
For example, if given a center at \((3, -5)\) with a radius \(8\), you will:

1. Substitute the center coordinates. Replace \(h\) with \(3\) and \(k\) with \(-5\), modifying the equation to \((x - 3)^2 + (y + 5)^2 = r^2\).
2. Substitute the radius. Plug \(r = 8\) into the equation. Since \(r^2\) equals \(64\), the full equation becomes \((x - 3)^2 + (y + 5)^2 = 64\).

This final equation now completely characterizes the circle in mathematical form, making it easy to graph and understand on a coordinate plane.