Understanding the Equation of a Circle Centered at the Origin

When dealing with geometric shapes and equations, the circle is a fundamental figure. This article explores the specific case of a circle centered at the origin (0, 0) and passing through a given point. The process of determining the necessary equation involves a few key steps based on the Pythagorean theorem and basic algebra.

Introduction to the Problem

The task is to find the equation of a circle situated at the origin and that passes through a particular point, specifically the point (1, -1).

Step-by-Step Solution

To find the equation of a circle centered at the origin and passing through a given point, we start with the general form of the circle's equation:

x^2 y^2 r^2

Here, r represents the radius of the circle. When the circle is centered at the origin (0, 0), the equation simplifies to this form. To determine the radius, we use the distance from the origin to the given point (1, -1). This distance is essentially the radius of the circle.

The distance formula derived from the Pythagorean theorem is:

r^2 x^2 y^2

Step 1: Calculate the Radius

Substitute the coordinates of the point (1, -1) into the equation:

r^2 1^2 (-1)^2

1^2 (-1)^2 1 1 2

Hence, r^2 2. Taking the square root of both sides, we find the radius r:

r sqrt{2}

Step 2: Formulate the Equation

Now that we have the radius, we can substitute it back into the general equation of the circle:

x^2 y^2 2

This equation describes the circle centered at the origin with a radius of sqrt{2}.

Conclusion

In summary, the equation of the circle centered at the origin (0, 0) and passing through the point (1, -1) is:

x^2 y^2 2

Understanding this process is crucial for solving similar geometric problems involving circles and their equations. The circle's properties and the Pythagorean theorem are key tools in this analytical approach.

Potential Further Exploration

For a deeper dive, one might explore the general form of the circle's equation for different centers and points, or look into more complex geometric shapes and their equations. This foundational knowledge can serve as a stepping stone for advanced topics in mathematics, such as conic sections and coordinate geometry.