To trace the curve represented by the equation:

$y(x^2 + 4a^2) = 8a^3$

we can follow these steps:

Step 1: Simplify the equation

Rewrite the equation in terms of $y$:

$y = \frac{8a^3}{x^2 + 4a^2}$

This form will help us understand the behavior of $y$ as a function of $x$.

Step 2: Analyze special points and symmetry

1. When $x = 0$: $y = \frac{8a^3}{0^2 + 4a^2} = \frac{8a^3}{4a^2} = 2a$ So, the point $(0, 2a)$ is on the curve.

2. As $x \to \infty$ or $x \to -\infty$: $y = \frac{8a^3}{x^2 + 4a^2} \to 0$ Thus, the curve approaches the $x$-axis as $x$ becomes large in either direction.

3. When $x = \pm 2a$: $y = \frac{8a^3}{(2a)^2 + 4a^2} = \frac{8a^3}{4a^2 + 4a^2} = \frac{8a^3}{8a^2} = a$ So, the points $(2a, a)$ and $(-2a, a)$ are also on the curve.

Step 3: Symmetry

Since $y$ only depends on $x^2$, the curve is symmetric about the $y$-axis. This means that for every point $(x, y)$ on the curve, there is a corresponding point $(-x, y)$.

Step 4: Behavior near the origin

Around $x = 0$, the function is well-behaved and takes a finite value, $y = 2a$.

Step 5: Asymptotic behavior

As $|x| \to \infty$, the function approaches the $x$-axis (i.e., $y = 0$) asymptotically. This suggests that the curve flattens out far from the origin.

Step 6: Final curve sketch and interpretation

The curve starts at $(0, 2a)$, descends symmetrically as $|x|$ increases, and asymptotically approaches the $x$-axis. It passes through the points $(2a, a)$ and $(-2a, a)$, and flattens out as $x \to \infty$ or $x \to -\infty$.

Conclusion:

The curve is symmetric about the $y$-axis, starts at $y = 2a$ for $x = 0$, passes through the points $(2a, a)$ and $(-2a, a)$, and approaches the $x$-axis as $|x|$ increases.

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Do you want to discuss any specific part of this tracing in more detail?

Here are 5 related questions to expand on the concepts:

1. How do you find the intercepts of a curve in general?
2. What is the significance of asymptotes in curve sketching?
3. How do you analyze the behavior of a curve at infinity?
4. How does symmetry simplify the tracing of curves?
5. How do singular points affect the shape of a curve?

Tip: When sketching curves, always check for symmetry, intercepts, and asymptotic behavior to simplify your analysis.