%5E2%2B(y-3)%5E2%3D25.jpeg "(x+2)^2+(y-3)^2=25")
Understanding the Equation (x+2)^2 + (y-3)^2 = 25
The equation (x+2)^2 + (y-3)^2 = 25 represents a circle in the coordinate plane. Let's break down why:
The Standard Form of a Circle
The general equation for a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Analyzing our Equation
By comparing our given equation with the standard form, we can identify the key features:
- Center: The center of the circle is at (-2, 3). Notice the signs are opposite in the equation (x + 2) and (y - 3).
- Radius: The radius of the circle is 5. This is because 25 is the square of the radius (r^2 = 25, so r = 5).
Visualizing the Circle
To graph the circle, we can follow these steps:
- Plot the center: Locate the point (-2, 3) on the coordinate plane.
- Radius points: Since the radius is 5, move 5 units to the right, left, up, and down from the center to find four points on the circle.
- Connect the points: Draw a smooth curve connecting the points to form the circle.
Key Takeaways
Understanding the standard form of a circle equation allows us to quickly identify the center and radius, making it easy to visualize and graph the circle. The equation (x+2)^2 + (y-3)^2 = 25 defines a circle with a center at (-2, 3) and a radius of 5 units.