(x-2)^2+(y+3)^2=36

3 min read Jun 17, 2024

Exploring the Equation: (x-2)^2 + (y+3)^2 = 36

The equation (x-2)^2 + (y+3)^2 = 36 represents a circle in the Cartesian coordinate plane. Let's break down why and explore its key features:

Understanding the Standard Form

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

Comparing this to our given equation, we can easily identify the following:

Center: (2, -3)
Radius: 6 (since the square root of 36 is 6)

Visualizing the Circle

Now that we know the center and radius, we can visualize the circle on a graph.

Plot the center: Locate the point (2, -3) on the coordinate plane.
Draw the radius: From the center, mark points 6 units away in all directions (up, down, left, right). These points will lie on the circumference of the circle.
Connect the points: Draw a smooth curve connecting the points to form the circle.

Key Properties

Symmetry: Circles are symmetrical around their center. Any line passing through the center divides the circle into two equal halves.
Circumference: The total distance around the circle is given by the formula C = 2πr, where r is the radius.
Area: The region enclosed by the circle is given by the formula A = πr^2, where r is the radius.

Conclusion

The equation (x-2)^2 + (y+3)^2 = 36 represents a circle centered at (2, -3) with a radius of 6. By understanding the standard form of the equation and its properties, we can easily visualize and analyze the characteristics of this geometric shape.