(x^2+y^2)^2=a(x^2-y^2) Graph

Jasonbradley

5 min read

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Jun 17, 2024

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Exploring the Graph of (x^2 + y^2)^2 = a(x^2 - y^2)

The equation (x^2 + y^2)^2 = a(x^2 - y^2), where a is a constant, describes a fascinating family of curves with unique characteristics. Let's delve into understanding its graph and explore its properties.

Analyzing the Equation

The equation is a quartic equation, meaning it involves terms with the highest power of 4. This already hints at a more complex graph than simple linear or quadratic functions.

To visualize the graph, we can start by rearranging the equation and making a few observations:

1. Symmetry: The equation remains unchanged if we substitute x with -x or y with -y. This signifies that the graph is symmetrical about both the x-axis and y-axis.

2. Polar Coordinates: Converting to polar coordinates (x = r cos θ, y = r sin θ) can simplify the equation:

   (r^2)^2 = a(r^2 cos^2 θ - r^2 sin^2 θ) r^4 = ar^2 (cos^2 θ - sin^2 θ) r^2 = a (cos^2 θ - sin^2 θ)

   This form suggests that the graph's shape depends on the value of a.

3. Special Cases:

   • a = 0: The equation becomes (x^2 + y^2)^2 = 0, which implies x = 0 and y = 0. The graph reduces to a single point at the origin.
   • a > 0: The equation results in a curve that resembles a lemniscate (a figure-eight shape). The size and orientation of the lemniscate depend on the value of a.
   • a < 0: This case results in a "pseudo-lemniscate" where the curve doesn't close, creating an open shape with two symmetrical loops.

Graphing the Curve

To get a better understanding, let's consider a few examples with different values of a:

• a = 1: The equation becomes (x^2 + y^2)^2 = (x^2 - y^2). The graph is a lemniscate with its loops oriented along the x-axis.

• a = -1: The equation becomes (x^2 + y^2)^2 = -(x^2 - y^2). The graph forms a "pseudo-lemniscate" with two open loops extending along the x-axis.

• a = 4: The equation becomes (x^2 + y^2)^2 = 4(x^2 - y^2). The graph is a larger lemniscate compared to the case when a = 1.

By plotting these and other cases, you'll observe how the shape and size of the curves evolve with varying values of a.

Conclusion

The equation (x^2 + y^2)^2 = a(x^2 - y^2) defines a family of interesting curves with diverse shapes. The parameter a plays a crucial role in determining the type of curve (lemniscate or pseudo-lemniscate) and its overall dimensions. This equation provides a fascinating glimpse into the world of complex equations and their visual representations.