%5E2%2B(y%2B3)%5E2%3D25.jpeg "(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:
- 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.