Problem 40 - Algebra 2

Question

Graph each circle. $$ (x-2)^{2}+(y+1)^{2}=36 $$

Step-by-Step Solution

Answer

The circle's center is at the point (2, -1) and the radius is 6 units.

Step 1: Identify the center

The center of a circle in the standard form \((x-h)^2 + (y-k)^2 = r^2\) is at the point (h, k). Looking at the equation \((x-2)^2 + (y+1)^2 = 36\), we see that h = 2 and k = -1. Thus, the center is at the point (2, -1).

Step 2: Identify the radius

The radius of a circle in the standard form \((x-h)^2 + (y-k)^2 = r^2\) is given by r. Looking at the equation, we have \(r^2 = 36\), so, solving for 'r', we find that 'r' = 6.

Step 3: Graph the circle

To graph the circle, first, plot the center point (2, -1) on the plane. Then draw a circle with radius 6 units around this point. You can also plot additional points by moving 6 units in all directions from the center to make sure the circle is round.

Key Concepts

Circle Equation

The equation of a circle in its standard form is written as \[(x-h)^2 + (y-k)^2 = r^2\].This formulation represents a circle on the Cartesian plane where:

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

It's important to notice this structure because identifying the specific parts of the equation allows us to easily locate the circle's center and determine its size.
In the equation provided from the exercise, \[(x-2)^2 + (y+1)^2 = 36\],we see that it matches the standard form exactly, making it straightforward to extract the necessary information about the circle.

Finding Center of a Circle

To determine the center of a circle represented by the equation \((x-h)^2 + (y-k)^2 = r^2\),one simply needs to identify the values \(h\) and \(k\) within the equation. These values are directly derived from the terms \((x-h)\) and \((y-k)\).
From the specific circle equation \((x-2)^2 + (y+1)^2 = 36\):

• The term \(x-2\) suggests that \(h = 2\).
• The term \(y+1\) implies \(k = -1\).

Hence, the center of the circle is located at the point \((2, -1)\).Understanding how to extract the center from the equation helps in plotting the circle accurately on a graph, ensuring it is placed correctly with respect to the x and y axes.

Radius of a Circle

The radius of a circle is a measure of its size. It is the distance from any point on the circle to its center.
In the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\),\(r^2\) represents the square of the radius. Thus, to find the radius, we need to take the square root of both sides of the equation.
Examining the given equation from the exercise \((x-2)^2 + (y+1)^2 = 36\), we see that \(r^2 = 36\).Taking the square root results in \(r = 6\).

• This means the circle extends 6 units from its center in all directions.
• The radius is crucial for drawing the circle on a graph, ensuring its round and symmetrical shape.