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The Meaning of (x² + y² - 1)x²y² = 0
The equation (x² + y² - 1)x²y² = 0 is a algebraic equation that represents a circle in the complex plane. In this article, we will break down the equation and explain what it represents.
Understanding the Equation
The equation can be factorized into two parts:
(x² + y² - 1) = 0 ... (1) x²y² = 0 ... (2)
Let's analyze each part separately:
Part 1: (x² + y² - 1) = 0
This equation represents a unit circle centered at the origin (0, 0) in the complex plane. The equation can be rewritten as:
x² + y² = 1
This is the standard equation of a circle with radius 1 and center at the origin.
Part 2: x²y² = 0
This equation represents the intersection of the x-axis and the y-axis in the complex plane. In other words, it represents the points where either x or y (or both) are equal to zero.
Geometric Interpretation
The intersection of the two equations represents the points where the unit circle intersects the x-axis and the y-axis. These points are:
- (1, 0)
- (-1, 0)
- (0, 1)
- (0, -1)
These points are the solutions to the equation (x² + y² - 1)x²y² = 0.
Conclusion
In conclusion, the equation (x² + y² - 1)x²y² = 0 represents the intersection of a unit circle and the x-axis and y-axis in the complex plane. The solutions to the equation are the points where the circle intersects the axes.
This equation is an important concept in mathematics, particularly in algebra and geometry. It has many applications in fields such as physics, engineering, and computer science.
