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:

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 xaxis and yaxis.

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.

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 figureeight shape). The size and orientation of the lemniscate depend on the value of a.
 a < 0: This case results in a "pseudolemniscate" 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 xaxis.

a = 1: The equation becomes (x^2 + y^2)^2 = (x^2  y^2). The graph forms a "pseudolemniscate" with two open loops extending along the xaxis.

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 pseudolemniscate) and its overall dimensions. This equation provides a fascinating glimpse into the world of complex equations and their visual representations.