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

4 min read Jun 17, 2024

Understanding the Circle Equation: (x-h)^2 + (y-k)^2 = r^2

The equation (x-h)^2 + (y-k)^2 = r^2 represents the standard form of a circle's equation. This equation provides a concise way to describe a circle's location and size in the coordinate plane.

Here's a breakdown of what each component signifies:

(x, y): Represents any point on the circle.
(h, k): Represents the center of the circle.
r: Represents the radius of the circle.

Example 1: Finding the Center and Radius

Let's say we have the equation (x - 3)^2 + (y + 2)^2 = 16.

Center: The center of the circle is at (h, k) = (3, -2).
Radius: The radius of the circle is √16 = 4.

Example 2: Writing the Equation from Given Information

Imagine we know the center of a circle is at (1, 5) and the radius is 3.

We can directly plug these values into the standard equation:

(x - 1)^2 + (y - 5)^2 = 3^2

This simplifies to (x - 1)^2 + (y - 5)^2 = 9.

Example 3: Finding the Equation from a Graph

Let's say we have a circle plotted on a graph with its center at (-2, 1) and it passes through the point (0, 1).

Find the radius: The distance between the center (-2, 1) and the point (0, 1) is 2 units. So, the radius is r = 2.
Write the equation: Using the center (h, k) = (-2, 1) and the radius r = 2, we get:

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

This simplifies to (x + 2)^2 + (y - 1)^2 = 4.

Applications of the Circle Equation

The equation (x-h)^2 + (y-k)^2 = r^2 has numerous applications in various fields, including:

Geometry: Determining the properties of circles, such as their area, circumference, and relationships with other geometric shapes.
Physics: Describing circular motion, such as the path of a satellite orbiting Earth.
Computer Graphics: Creating circular objects and effects in computer programs and games.

By understanding the components and applications of the circle equation, you gain a powerful tool for solving problems related to circles and understanding their role in various areas of science and technology.