The Circle Equation: (x3)^2 + (y+2)^2 = 4
The equation (x3)^2 + (y+2)^2 = 4 represents a circle in the Cartesian coordinate system. Let's explore the characteristics of this circle and how to interpret its equation.
Understanding the Standard Circle Equation
The general form of the equation for a circle is:
(x  h)^2 + (y  k)^2 = r^2
Where:
 (h, k) represents the coordinates of the center of the circle.
 r represents the radius of the circle.
Analyzing the Given Equation
In our equation, (x3)^2 + (y+2)^2 = 4, we can identify the following:

Center: The center of the circle is located at (3, 2). This is because the equation is in the form (x  h)^2 + (y  k)^2 = r^2, where h = 3 and k = 2.

Radius: The radius of the circle is 2. We obtain this by taking the square root of the constant term on the right side of the equation, which is 4. Therefore, r = √4 = 2.
Graphing the Circle
To visualize this circle, we can plot the center point (3, 2) and then use the radius of 2 to locate points on the circle.

Start at the center (3, 2).

Move 2 units to the right, left, up, and down from the center. These points will be on the circle.

Connect these points smoothly to form the circle.
Summary
The equation (x3)^2 + (y+2)^2 = 4 represents a circle centered at (3, 2) with a radius of 2. By understanding the standard circle equation and analyzing its components, we can easily determine the characteristics and visualize the circle represented by any given equation.