Problem 21 - Algebra A Combined Function

Question

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ (x-5)^{2}+(y+2)^{2}=1 $$

Step-by-Step Solution

Answer

Center: (5, -2), Radius: 1

Step 1: Identify the standard form of a circle's equation

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

Step 2: Extract the center from the equation

From the given equation \((x-5)^2 + (y+2)^2 = 1\), compare it to the standard form. We can see that \(h = 5\) and \(k = -2\). Therefore, the center is \((5, -2)\).

Step 3: Determine the radius from the equation

The given equation can be written in the form \((x-5)^2 + (y+2)^2 = 1\), where \(r^2 = 1\). Solving for \(r\), we get \(r = \sqrt{1} = 1\). Thus, the radius is 1.

Step 4: Verify and prepare to graph the circle

With the center \((5, -2)\) and radius 1, you can plot the circle on the coordinate plane. The circle's edge will be anywhere 1 unit away from the center.

Key Concepts

Center of a Circle

To find the center of a circle from its equation, it's important to recognize the formula in standard form. The equation given in the problem resembles \[(x-h)^2 + (y-k)^2 = r^2,\]where

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

This means the center of the circle is the point \((h, k)\). To identify these from the equation \((x-5)^2 + (y+2)^2 = 1\), compare it directly with the standard form:
The values from the equation \((x-5)^2 + (y+2)^2 = 1\) show that \(h=5\) and \(k=-2\). So, the center is at the point \((5, -2)\) on the coordinate plane.
This precise identification of the center is crucial for graphing the circle, as the entire graph is based around this central point.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. It's a crucial property that determines the size of the circle.
To find the radius from the standard circle equation:\((x-h)^2 + (y-k)^2 = r^2\), we look at \(r^2\), the portion of the equation after the equals sign.
For the given equation, \((x-5)^2 + (y+2)^2 = 1\), this becomes \(r^2 = 1\).
This implies that:

• To solve for \(r\), simply calculate the square root of 1, giving \(r=1\).

So, the circle's radius is 1 unit.
Knowing the radius is just as essential as knowing the center because it tells exactly how far the circle extends in every direction from the center point.

Standard Form of a Circle Equation

Understanding the standard form of a circle equation is key to analyzing and graphing circles. This standard form is: \((x-h)^2 + (y-k)^2 = r^2\).
This formula is particularly structured to make identifying the circle's center and radius straightforward:

• The expressions \((x-h)^2\) and \((y-k)^2\) represent the squared distances from \(x\) and \(y\) to their central coordinates, \(h\) and \(k\), respectively.
• Solving for \(r\) gives the actual radius, by taking the square root of the equation's right-hand side.

With this form, you can easily extract the circle’s essential characteristics directly from the equation.
For example, converting a given circle equation into this form—for instance, \((x-5)^2 + (y+2)^2 = 1\)—confirms it's already in standard form and helps us identify that the center is \((5, -2)\) with a radius of 1.
This makes it handy for not just solving problems but also graphing circles seamlessly.