Graphing the Equation (x^2y^2 - 2x 0): A Complete Guide

In this article, we will explore the process of graphing the equation (x^2y^2 - 2x 0). We will rewrite the equation in standard form, identify its center and radius, and plot the resulting circle. Understanding these steps will provide a clear and comprehensive view of the geometric properties of the equation.

Rewriting the Equation

The given equation is:

[x^2y^2 - 2x 0]

First, let's isolate the terms involving (x) and complete the square.

Completing the Square

We start by adding and subtracting the same value to complete the square:

[x^2 - 2x 1 - 1 y^2 0]

This simplifies to:

[x^2 - 2x 1 y^2 1]

And further simplifies to:

[(x - 1)^2 y^2 1]

Identifying the Circle's Properties

The equation [(x - 1)^2 y^2 1] represents a circle in the standard form of the equation of a circle: ((x - h)^2 (y - k)^2 r^2).

Center: The center of the circle is at the point ((1, 0)).Radius: The radius of the circle is (1).

Plotting the Circle

To plot the circle, follow these steps:

Plotting Hints

Start by plotting the center ((1, 0)).Since the radius is (1), the circle will extend 1 unit in all directions from the center. Therefore, the points ((0, 0)) and ((2, 0)) are on the horizontal diameter, and the points ((1, 1)) and ((1, -1)) are on the vertical diameter.

Sketching the Graph

Here is a quick sketch of the graph:

Graph of ((x - 1)^2 y^2 1)

In this graph, the x-axis is denoted as (x), and the y-axis is denoted as (y). The center of the circle is marked as (A), and the points on the circle are marked as (B) and (C).

Further Exploration

The given equation can also be viewed as a sum of squares:

[x^2 - 2x y^2 0 1 - 1]

This simplifies to:

[(x - 1)^2 y^2 1]

This confirms that the equation represents a circle with center ((1, 0)) and radius (1).

Conclusion

By rewriting the equation (x^2y^2 - 2x 0) in the standard circle form, we can identify its center and radius. The circle is centered at ((1, 0)) with a radius of (1).

To summarize, the equation (x^2y^2 - 2x 0) can be graphed by plotting the center at ((1, 0)) and extending a radius of (1) unit in all directions.

Keywords: graphing equations, center-radius form, completing the square