Reflecting a Circle Across the X-Axis: A Comprehensive Guide

The task of reflecting a circle across the X-axis is a fundamental concept in geometry and algebra. This guide will explain the process step-by-step, illustrating how the original circle's equation changes under such a transformation. We'll start with the original circle's equation and then derive the equation for the reflected circle, providing a detailed explanation along with a graphical solution for better understanding.

Original Circle Equation and Its Properties

The equation of the original circle is given as:

x - 3^2 y - 4^2 25

This equation represents a circle with its center at the point (3, 4) and a radius of 5. This is determined by rewriting the equation in standard form, (x - h)^2 (y - k)^2 r^2, where ((h, k) is the center and r is the radius. For our equation, we can identify h 3, k 4, and r 5.

Reflecting the Circle Across the X-Axis

When reflecting a point across the X-axis, the y-coordinate changes sign. Therefore, the center of the circle at (3, 4) will be reflected to (3, -4). The formula for reflecting a point across the X-axis is to negate the value of the y-coordinate and keep the x-coordinate the same.

Deriving the Equation of the Reflected Circle

To derive the equation of the reflected circle, we substitute the new center coordinates into the standard form of the circle's equation:

(x - 3)^2 (y 4)^2 25

This equation represents the reflected circle with center at (3, -4) and a radius of 5. The process involves changing the y-coordinate of the original center by negating it. Thus, we have:

(x - 3)^2 (y - (-4))^2 25

Which simplifies to:

(x - 3)^2 (y 4)^2 25

Graphical Solution

A graphical solution can help visualize the transformation. The original circle has its center at (3, 4) and the reflected circle will have its center at (3, -4). Both have the same radius of 5 units.

Step-By-Step Reflection Process

Identify the original circle's center: (3, 4). Negate the y-coordinate of the center: (3, -4). Write the equation of the reflected circle using the new center: (x - 3)^2 (y 4)^2 25.

Standard Form of the Circle's Equation

Understanding the standard form of the equation of a circle, (x - a)^2 (y - b)^2 r^2, where (a, b) is the center and r is the radius, is crucial for solving these geometric transformations. Here, the center is transformed from (3, 4) to (3, -4) and the radius remains 5 units.

Summary and Conclusion

In summary, reflecting a circle across the X-axis involves changing the sign of the y-coordinate of the center while keeping the x-coordinate the same. The equation of the reflected circle can be written as:

(x - 3)^2 (y 4)^2 25

This guide has provided a comprehensive approach to solving such problems, ensuring that the center and radius of the circle are correctly identified and transformed.