Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the center and radius of the circle. $$x^{2}+(y+8)^{2}=25$$

Step-by-Step Solution

Answer

The center of the circle is at (0, -8) and the radius is 5.

Step 1: Identify Center

The center of the circle is at \((h, k)\). From the equation, we can infer that \(h = 0\) and \(k = -8\), as \(x^{2}\) would indicate \(h = 0\) and \((y+8)^{2}\) would indicate \(k = -8\). Therefore, the center of the circle is at \((0, -8)\).

Step 2: Identify Radius

The radius of the circle is \(r\), which is the square root of the right side of the equation. In this case, \(r^{2}=25\). Taking the square root of 25, we find that \(r = 5\). Therefore, the radius of the circle is 5.

Key Concepts

Center of a Circle

To find the center of a circle using its equation, we need to understand the standard form of a circle equation: \[(x - h)^2 + (y - k)^2 = r^2\]This equation neatly represents a circle where

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

In the given problem, the circle equation is \[x^2 + (y + 8)^2 = 25\]Notice that the equation does not have a \(h\) altering the \(x\) term. This means \(h = 0\). Also, the \((y + 8)\) term implies that \(k = -8\). Understanding these adjustments is key. - The value of \(h\) is the horizontal shift of the circle.- The value of \(k\) is the vertical shift of the circle.Inserting these values into the context of the original problem, the center of the circle is at the coordinates \( (0, -8) \). Identifying shifts effectively helps plot circles accurately on the coordinate plane.

Radius of a Circle

The radius is a fundamental part of the circle, helping define its size. In terms of a circle equation in its standard form: \[(x - h)^2 + (y - k)^2 = r^2\]The term \( r^2 \) on the right side is crucial to determine the circle's radius. To find the radius, simply take the square root of the constant on the right-hand side of the equation. For the problem at hand, the equation ends with \(25\), which means:- \( r^2 = 25\)To extract \( r \), compute the square root:\[r = \sqrt{25} = 5\]Thus, the radius of this circle is \(5\). This measurement is the distance from the center of the circle to any point on its circumference. Identifying this allows you to draw the circle to scale, whether on graph paper or using software. Understanding this ease calculation of area and circumference as well.

Standard Form of Circle Equation

The standard form of a circle equation is a pivotal tool for identifying key features of a circle's graph easily. This form is\[(x - h)^2 + (y - k)^2 = r^2\]Each component plays a specific role:

• The terms \( (x - h)^2 \) and \( (y - k)^2 \) translate the circle on the coordinate plane, adjusting its position based on \(h\) and \(k\) coordinates.
• The \( r^2 \) is fundamentally linked to the radius, dictating the circle's size.

For example, \[x^2 + (y + 8)^2 = 25 \]is a circle with a center located at \( (h, k) = (0, -8)\) and a radius \( r = 5 \). Recognizing this template helps transform complex problems into manageable tasks. When you learn to spot the standard form, modifications to the circle's position or size can be swiftly calculated by comparing given equations to this form. This aids in verifying solutions or graphing by hand or digitally.