Problem 41 - Algebra 2

Question

Graph each circle. $$ (x-1)^{2}+(y-1)^{2}=4 $$

Step-by-Step Solution

Answer

The graph of the circle will be centered at the point (1,1), with a radius of 2.

Step 1: Identify the center of the circle

The center of the circle (h,k) can be found by examining the terms in the equation. In this case, the center will be at the point (1,1).

Step 2: Identify the radius of the circle

The radius of the circle can be found by taking the square root of the number on the right side of the equation. In this case, the radius is \( \sqrt{4} \), which is 2.

Step 3: Graph the circle

Start by drawing a point at the center, (1,1). Then, measure out the radius (2) in all directions from the center and draw the circle. Ensure that the radius is the same length in every direction from the center.

Key Concepts

Center of Circle

The center of a circle is a crucial element that helps locate the exact position of the circle on a coordinate plane. In the standard equation of a circle, which is \[(x-h)^2 + (y-k)^2 = r^2\]any terms inside the brackets directly provide the center's coordinates. The center is represented as \((h, k)\), where:

• \(h\) is the x-coordinate of the center.
• \(k\) is the y-coordinate of the center.

For the circle \((x-1)^2 + (y-1)^2 = 4\), the center can be identified as \((1, 1)\):- Notice how the equation matches the standard form where \((x-h)^2\) and \((y-k)^2\) indicate the horizontal and vertical shifts of the circle's center from the origin.Understanding this helps you readily pinpoint the location of the circle in any graph.

Radius of Circle

The radius of a circle plays a central role in defining the size of the circle. It is the distance from the center of the circle to any point on its edge. In the circle's standard equation \((x-h)^2 + (y-k)^2 = r^2\), the term on the right side of the equation, \(r^2\), provides the square of the radius.To determine the actual radius, you take the square root of \(r^2\). Let's look at the specific example \((x-1)^2 + (y-1)^2 = 4\):

• The right side of the equation is 4, which equals \(r^2\).
• Find \(r\) by calculating \(\sqrt{4} = 2\).

This means the radius of the circle is 2. Knowing how to find the radius gives you the measure of how far out to extend from the center to draw the circle accurately.

Equation of a Circle

The equation of a circle is essential for graphing and understanding its geometric properties. The standard form \((x-h)^2 + (y-k)^2 = r^2\) contains all the necessary details to illustrate the circle:- The left side of the equation includes \((x-h)^2\) and \((y-k)^2\), allowing you to determine the center of the circle from \(h\) and \(k\).- The right side, depicted as \(r^2\), is used to calculate the radius, \(r\).Given the equation \((x-1)^2 + (y-1)^2 = 4\):

• The center is deduced to be \((1,1)\).
• The radius is found to be \(2\), after taking the square root of \(4\).

Understanding this form is instrumentally helpful because it provides a precise method to graph circles and allows you to comprehend their sizes and locations effortlessly.