(x-1)^2+(y-2)^2=4

3 min read Jun 17, 2024

Exploring the Circle: (x-1)^2 + (y-2)^2 = 4

The equation (x-1)^2 + (y-2)^2 = 4 represents a circle in the coordinate plane. This article will delve into understanding the properties of this circle and how its equation reveals its key characteristics.

The Standard Form of a Circle Equation

The equation (x-1)^2 + (y-2)^2 = 4 is in the standard form of a circle equation:

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

Where:

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

Identifying the Center and Radius

Comparing the given equation to the standard form, we can identify the center and radius:

Center: (h, k) = (1, 2)
Radius: r^2 = 4, therefore r = 2

Visualizing the Circle

To visualize the circle, we can plot the center point (1, 2) and draw a circle with a radius of 2 units around it. The circle will pass through points that are 2 units away from the center in all directions.

Understanding the Equation

The equation itself represents the Pythagorean theorem applied to every point on the circle. Consider a point (x, y) on the circle. The distance between this point and the center (1, 2) is given by:

√((x - 1)^2 + (y - 2)^2)

Since this distance is equal to the radius (2), we have:

√((x - 1)^2 + (y - 2)^2) = 2

Squaring both sides gives us the original equation:

(x - 1)^2 + (y - 2)^2 = 4

Conclusion

The equation (x-1)^2 + (y-2)^2 = 4 concisely describes a circle with a center at (1, 2) and a radius of 2 units. By understanding the standard form of a circle equation and the underlying concepts, we can easily identify the key features of a circle from its equation and visualize its position on the coordinate plane.