Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the equation of a circle with the given center and radius. Graph the circle. Center \((0,4),\) radius \(\sqrt{6}\)

Step-by-Step Solution

Answer

The center-radius form of the circle's equation is \\x^2 + (y-4)^2 = 6\\.

Step 1: Identify the Standard Equation of a Circle

The standard equation of a circle with center at \(h, k\) and radius \r\ is given by \( (x-h)^2 + (y-k)^2 = r^2 \). In this exercise, we are provided the center and the radius to apply this formula.

Step 2: Substitute Known Values into the Equation

Given in the problem, the center of the circle is \(0, 4\) and the radius is \sqrt{6}\. Substitute this into the standard equation: \( (x-0)^2 + (y-4)^2 = (\sqrt{6})^2 \).

Step 3: Simplify the Equation

We simplify the equation by evaluating \( (x-0)^2 \) as \( x^2 \), and \((\sqrt{6})^2\) as 6. This transforms the equation into: \ x^2 + (y-4)^2 = 6 \.

Step 4: Graph the Circle

Use the center \(0, 4\) to locate the center of the circle on the graph. The radius \sqrt{6}\ can be visualized by plotting points at a distance \sqrt{6}\ from the center in all directions, and then sketching a circle through these points.

Key Concepts

Standard Form of a Circle

Understanding the standard form of a circle's equation is essential when working with problems involving circles. This equation, given by \((x-h)^2 + (y-k)^2 = r^2\), represents a circle in a specific location on the Cartesian coordinate plane.

• \((h, k)\) denotes the center of the circle.
• \(r\) stands for the radius.

This form is extremely helpful because it clearly shows both the center and radius of the circle, making it straightforward to graph or manipulate.
It allows one to easily identify or verify any circle's position and size with only a quick glance at the equation parts.

Graphing Circles

Graphing a circle involves a simple yet methodical process. The first step is identifying the center of the circle. In our example, it's located at \((0, 4)\). This center is the point on your graph where you'll begin.
Once you have the center, you use the radius, \(\sqrt{6}\) in this case, to determine how far from that center the edge of the circle will reach. Visualize or calculate this distance in all directions—up, down, left, right, and diagonally—to accurately sketch the circle's boundary.
This method effectively translates the equation into a visual representation.

Center-Radius Form of a Circle

The center-radius form is a type of circle equation designed to make graphing and understanding circles easier. It clearly presents two crucial elements:

• The center, \((h, k)\).
• The radius, \(r\).

The equation structure \((x-h)^2 + (y-k)^2 = r^2\) simplifies figuring out where a circle is positioned and how large it is. From our example:

x^2 + (y-4)^2 = 6

This shows a circle centered at \((0, 4)\) with a radius of \(\sqrt{6}\). By using this form, you can effortlessly move between the equation and the graph, making it an effective tool for both learners and practitioners.