(x-1)^2+(y+3)^2=25

3 min read Jun 17, 2024
(x-1)^2+(y+3)^2=25

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:

  1. Plot the center: Mark the point (1, -3) on a coordinate plane.
  2. Draw the radius: From the center, draw lines in all directions with a length of 5 units.
  3. 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.