Problem 17 - Intermediate Algebra

Question

Find the center and radius of each circle and graph it. $$ x^{2}+(y+3)^{2}=1 $$

Step-by-Step Solution

Answer

Center: (0, -3), Radius: 1.

Step 1: Identify the Standard Form

The equation provided is in the standard form for a circle: \[(x-h)^2 + (y-k)^2 = r^2\] where \(h\) and \(k\) represent the coordinates of the center of the circle, and \(r\) is the radius of the circle.

Step 2: Determine the Center of the Circle

Compare the given equation \(x^2 + (y+3)^2 = 1\) with the standard form: \[(x-0)^2 + (y+(-3))^2 = 1\] It shows that the center of the circle is \((h, k) = (0, -3)\).

Step 3: Calculate the Radius of the Circle

The radius \(r\) is the square root of the constant on the right side of the equation. Since \(1\) is on the right side:\[r = \sqrt{1} = 1\]Thus, the radius of the circle is 1.

Key Concepts

Standard Form of a Circle

The standard form of a circle equation is a fundamental concept in geometry, allowing us to easily identify key properties such as the center and radius of a circle. It is expressed as:\[ (x-h)^2 + (y-k)^2 = r^2 \]Here,

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

This form is useful because it clearly displays the circle's geometric information. By comparing a given equation with this standard form, one can directly extract the circle's center and radius. This makes it a powerful tool for solving circle-related problems and is especially handy when graphing circles.

Center of a Circle

Determining the center of a circle from its equation is straightforward when using the standard form. In the context of the equation \(x^2 + (y+3)^2 = 1\), it aligns with the standard form of:\[ (x-h)^2 + (y-k)^2 = r^2 \]By observation, we can see:

• \(h = 0\)
• \(k = -3\)

Thus, the center of the circle is \((0, -3)\). Understanding the center is crucial as it serves as a reference point for the circle's position on the coordinate plane. It tells us exactly where the circle is located relative to the origin.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle itself. It is found by taking the square root of the constant on the right side of the standard form equation. In the equation provided, \(x^2 + (y+3)^2 = 1\), the constant is 1.Calculating the radius involves extracting the square root from this constant value:

• \(r = \sqrt{1} = 1\)

Thus, the radius of this circle is 1. The radius helps determine the size of the circle and can assist in graphing, as it dictates how far each point on the boundary is from the center.

Graphing Circles

Graphing a circle involves plotting its center on a coordinate plane and then using its radius to draw the boundary. For the circle described by the equation \(x^2 + (y+3)^2 = 1\), we have already identified:

• The center at \((0, -3)\)
• The radius of 1

To graph this circle:1. Plot the center point \((0, -3)\) on the coordinate plane.2. Use a compass or draw points at a distance of 1 unit from the center in all directions.3. Connect these points smoothly to outline the circle.This visualization helps in understanding the circle's position and size relative to other features on the coordinate plane. Graphing is a useful skill in mathematics that aids in visual comprehension of equations and data.