Problem 61 - Algebra 2

Question

Find the center and the radius of each circle. $$ (x+3)^{2}+(y-5)^{2}=38 $$

Step-by-Step Solution

Answer

The center of the circle is at (-3, 5) and the radius of the circle is \( \sqrt{38} \) units.

Step 1: Identify a, b and r^2

The circle's equation \( (x+3)^2 + (y-5)^2 = 38 \) can be compared to the standard form \( (x-a)^2 + (y-b)^2 = r^2 \). Here, a = -3 (opposite of +3), b = 5 (opposite of -5), and r^2 = 38.

Step 2: Find the radii

The square root of \( r^2 = 38 \) gives r. That is, \( r = \sqrt{38} \)

Key Concepts

Center of a Circle

Understanding the center of a circle is crucial when working with circle equations.
Each circle can be described by its equation in the standard form, and from this form, the center can be easily identified.
In the equation \[ (x-a)^2 + (y-b)^2 = r^2 \], the center is given by the point \((a, b)\).
This means it is located at the coordinates determined by \(a\) and \(b\).

• In a circle equation like \((x+3)^2 + (y-5)^2 = 38\), compare it with the standard form.
  
• The subtraction sign in the standard form gives us the actual coordinates by flipping the signs.
  

Thus, our circle's equation \((x+3)^2 + (y-5)^2 = 38\) has a center at \((-3, 5)\). The signs are inverted: +3 becomes -3, and -5 stays 5.
This is how you determine the center from a circle's equation.

Radius of a Circle

Once you know the center, finding the radius of a circle involves just a few simple steps.
The radius is the distance from the center to any point on the circle. In the standard form \((x-a)^2 + (y-b)^2 = r^2\), \(r^2\) represents the square of the radius.
To find the actual radius, take the square root of \(r^2\).

• In our example: \((x+3)^2 + (y-5)^2 = 38\), we see \(r^2 = 38\).
  
• The radius \(r\) is then given by \( \sqrt{38} \).
  

Approximately, \( \sqrt{38} \approx 6.16\), which means the circle's radius is about 6.16 units.
The radius gives you a sense of the size of the circle, showing how far, in units, the boundary of the circle is from the center.

Standard Form of a Circle

The standard form of a circle equation is \((x-a)^2 + (y-b)^2 = r^2\). This form is extremely helpful as it allows you to directly identify the circle's center and radius.
Let's break down what each component means:

• \((x-a)^2\): This part determines the horizontal position, and \(a\) is the x-coordinate of the center.
  
• \((y-b)^2\): This part denotes the vertical position, and \(b\) is the y-coordinate of the center.
  
• \(r^2\): This represents the square of the radius, giving you information about the size of the circle.
  

So, a circle in this form clearly provides both the center, \((a, b)\), and the radius, \( \sqrt{r^2}\).
By rewriting any circle equation into this form, you can easily extract these key details, simplifying the process of graphing or analyzing the circle.
This form is simple yet powerful, making it a favorite for learning and solving geometrical problems efficiently.