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.

Explore how to trace the curve defined by y(x^2 + 4a^2) = 8a^3. This problem highlights key concepts such as symmetry, asymptotic behavior, and special point analysis, suitable for students in Grades 11-12.

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