Problem 42 - Algebra A Combined Function

Question

Write an equation of the circle with the given center and radius. See Example 8. $$ (0,-6) ; \sqrt{2} $$

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + (y + 6)^2 = 2 \).

Step 1: Understand the General Equation of a Circle

The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] Identify the components needed: the center \( (h, k) \) and the radius \( r \).

Step 2: Identify the Center and Radius

From the problem, the center of the circle is given as \( (0, -6) \), which means \( h = 0 \) and \( k = -6 \). The radius is given as \( r = \sqrt{2} \).

Step 3: Substitute into the General Equation

Substitute \( h = 0 \), \( k = -6 \), and \( r = \sqrt{2} \) into the general circle equation:\[ (x - 0)^2 + (y + 6)^2 = (\sqrt{2})^2 \]

Step 4: Simplify the Equation

Simplify the equation from Step 3:\[ x^2 + (y + 6)^2 = 2 \]Here we simply squared the radius since \( (\sqrt{2})^2 = 2 \).

Key Concepts

General Equation of a Circle

The general equation of a circle provides a formulaic way to represent all points that are equidistant from a specific point, known as the center of the circle. The standard form of the circle's equation is \[(x - h)^2 + (y - k)^2 = r^2\]where:

• (h, k) is the center of the circle.
• r is the radius.

In this equation, (x, y) represents any point on the circumference of the circle. The expression \((x - h)^2 + (y - k)^2\) calculates the distance squared from any point on the circle to its center. This formula is derived from the Pythagorean theorem, which captures the essence of a circle as a set of points fixed at a constant distance (the radius) from a central point.

Center of a Circle

The center of a circle is a crucial part of its geometric definition. It is the point from which every point on the circle is equidistant. In the general equation \[(x - h)^2 + (y - k)^2 = r^2\]the center is represented by the coordinates \((h, k)\). Knowing the center is fundamental in identifying the exact position of the circle in a coordinate plane. For example, in the problem, the circle has a center of \((0, -6)\). This means the circle is vertically centered 6 units below the x-axis, and it aligns with the y-axis horizontally. Understanding the center helps in visualizing the circle’s location and for drawing it accurately on the graph.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. In mathematical terms, the radius is a non-negative number \(r\) in the circle equation, represented by \[(x - h)^2 + (y - k)^2 = r^2\].The radius defines the size of the circle. It is crucial because it determines how large or small the circle is. For instance, in this exercise, the radius is given as \(\sqrt{2}\). By substituting \(r = \sqrt{2}\) into the circle's equation, we can calculate the radius squared, which in this case results in 2 since \((\sqrt{2})^2 = 2\). Recognizing how to work with the radius allows us to fully describe the geometric attributes of the circle.