Problem 19 - Algebra 2

Question

For each equation, find the center and radius of the circle. $$ (x-1)^{2}+(y-1)^{2}=1 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Center

The center of the circle is given by the values of h and k in the equation. Looking at the equation, h and k are denoted by the numbers that are being subtracted from x and y respectively. So, the center of the circle is (h,k) = (1,1).

Step 2: Identify the Radius

The radius of the circle can be obtained directly from the equation, under the square root of the constant on the right side of the equation. Here, the constant is 1. So, the radius r is the square root of 1, which equals to 1.

Key Concepts

Center of a Circle

In the standard form of a circle’s equation, which is \((x-h)^2 + (y-k)^2 = r^2\), the center of the circle is represented by the coordinates \((h, k)\). These correspond to the numbers that are subtracted from \(x\) and \(y\) within the equation.
For example, in the equation \((x-1)^2 + (y-1)^2 = 1\), you can easily identify the center as \((1, 1)\). This is because \(h\) and \(k\) are both 1. Thus, the center is a crucial starting point when working with circle equations.

It's important to pinpoint the center correctly:

• Look for the terms inside the parentheses.
• The terms being subtracted give you \(h\) and \(k\).

This step ensures you know the exact location of the circle on the coordinate plane.

Radius of a Circle

The radius of a circle in the equation \((x-h)^2 + (y-k)^2 = r^2\) is determined by the constant \(r^2\) on the right side. To find the radius, you take the square root of this constant.
In our example, the equation \((x-1)^2 + (y-1)^2 = 1\), the constant is 1. Thus, the radius \(r\) is \(\sqrt{1} = 1\).

Understanding the radius is simple if you follow these steps:

• Find the constant on the right side of the equation.
• Take the square root to determine \(r\).

The radius defines the size of the circle and how far its edge is from the center.

Standard Form of a Circle

The standard form of a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\). It neatly encapsulates the circle's geometry in a simple expression.
This form makes it easy to recognize both the center and radius of the circle.
For a clear understanding, consider this:

• \(h\) and \(k\) in \((x-h)^2 + (y-k)^2\) represent the circle’s center.
• \(r^2\) is the square of the radius.

Using this formula helps in sketching and analyzing circles effectively on a graph.
It also provides a consistent way to describe circles, making it an essential tool in algebra and geometry.