Problem 64 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ (x+1)^{2}+(y-1)^{2}=16 $$

Step-by-Step Solution

Answer

Center: \((-1, 1)\), Radius: 4.

Step 1: Identify the Circle Equation Form

The given equation \((x+1)^2 + (y-1)^2 = 16\) is in the standard form of a circle equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Determine the Center

From the equation \((x+1)^2 + (y-1)^2 = 16\), compare it with \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h = -1\) and \(k = 1\), so the center of the circle is \((-1, 1)\).

Step 3: Calculate the Radius

The given equation \((x+1)^2 + (y-1)^2 = 16\) shows \(r^2 = 16\). To find \(r\), calculate \(r = \sqrt{16}\), which simplifies to \(r = 4\).

Step 4: Summary

The center of the circle is \((-1, 1)\) and the radius is 4.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is crucial to understand when working with circles in coordinate geometry. This equation is written as:

• \((x-h)^2 + (y-k)^2 = r^2\)

where:

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

In this format, all the essential properties of the circle are easily identifiable, such as the location of its center and its size through the radius.

This form is particularly helpful because it allows us to identify these properties at a glance, just by comparing the given equation to the standard form, making it a powerful tool in geometry problems.

Center of a Circle

Understanding the center of a circle is key when analyzing circle equations. When given the standard form of a circle \((x-h)^2 + (y-k)^2 = r^2\), the center is extracted as follows:

• The term \( (x-h) \) reveals the x-coordinate of the center as \( h\).
• The term \( (y-k) \) reveals the y-coordinate of the center as \( k \).

Therefore, the center of the circle is the point \((h, k)\). For instance, in the equation \((x+1)^2 + (y-1)^2 = 16\), the center can be found by recognizing that:

• \(h = -1\)
• \(k = 1\)

Thus, the center of the circle is at the coordinates \((-1, 1)\).

Remember, the signs of \(h\) and \(k\) are opposite in the equation compared to their values in the center, which is a common point of confusion that needs special attention.

Radius of a Circle

The radius is one of the fundamental aspects of a circle, signifying the distance from the center to any point on its circumference. In the equation's standard form, the radius is denoted as \(r\). From the form \((x-h)^2 + (y-k)^2 = r^2\), we see that:

• \(r^2\) represents the square of the radius.

Thus, to find the radius \(r\), you take the square root of \(r^2\). For example, in the provided equation \((x+1)^2 + (y-1)^2 = 16\), \(r^2 = 16\).

Solving for \(r\), we calculate:

• \(r = \sqrt{16}\), which simplifies to \(r = 4\).

This means the circle has a radius of 4. Understanding this process ensures you can quickly determine the size of any circle given in standard form.