## Understanding the Equation: (x-1)^2 + (y+3)^2 = 25

The equation (x-1)^2 + (y+3)^2 = 25 represents a **circle** in the **Cartesian coordinate system**. Let's break down why this is and explore its properties.

### The Standard Form of a Circle Equation

The general equation for a circle is:

**(x - h)^2 + (y - k)^2 = r^2**

Where:

**(h, k)**represents the**center**of the circle.**r**represents the**radius**of the circle.

### Analyzing Our Equation

Comparing our equation (x-1)^2 + (y+3)^2 = 25 with the standard form, we can identify the following:

**Center:**(1, -3)**Radius:**√25 = 5

This means the circle is centered at the point (1, -3) and has a radius of 5 units.

### Visualizing the Circle

To visualize the circle, you can:

**Plot the center:**Mark the point (1, -3) on a coordinate plane.**Draw the radius:**From the center, draw lines in all directions with a length of 5 units.**Connect the points:**Connect the endpoints of the radius lines to form a smooth circle.

### Key Properties

The equation (x-1)^2 + (y+3)^2 = 25 defines a circle with the following properties:

**Symmetry:**The circle is symmetrical about both the x-axis and the y-axis.**Circumference:**The circumference of the circle is 2πr = 10π units.**Area:**The area of the circle is πr^2 = 25π square units.

### Conclusion

Understanding the standard form of a circle equation allows you to quickly identify its center, radius, and other key properties. By analyzing the equation (x-1)^2 + (y+3)^2 = 25, we can visualize and understand the specific circle it defines.