The problem asks to discuss the symmetry and the asymptote of the curve $y^2 (a - x) = x^3$, where $a > 0$, in the Cartesian plane.

Solution:

1. Symmetry

The given equation is: $y^2 (a - x) = x^3.$

• Step 1: Check for symmetry about the x-axis. Substitute $y \to -y$ into the equation: $(-y)^2 (a - x) = x^3 \implies y^2 (a - x) = x^3.$ The equation remains unchanged, so the curve is symmetric about the x-axis.

• Step 2: Check for symmetry about the y-axis. Substitute $x \to -x$ into the equation: $y^2 (a - (-x)) = (-x)^3 \implies y^2 (a + x) = -x^3.$ The resulting equation is not equivalent to the original, so the curve is not symmetric about the y-axis.

• Conclusion on symmetry: The curve is symmetric about the x-axis only.

2. Asymptotes

The asymptotes are determined by examining the behavior of the curve as $x \to \infty$ or $x \to -\infty$.

• Case 1: As $x \to a$ from the left ($x < a$): When $x$ is near $a$, $a - x \to 0^+$. Then: $y^2 \approx \frac{x^3}{a - x} \to +\infty.$ This means the curve shoots upwards or downwards near $x = a$, but $x = a$ is not an asymptote because the curve does not approach a straight line.

• Case 2: As $x \to \infty$: When $x \to \infty$, $a - x \to -\infty$. Then: $y^2 (a - x) = x^3 \implies y^2 \sim -\frac{x^3}{x} = -x^2.$ Since $y^2$ cannot be negative, there are no real points as $x \to \infty$.

• Case 3: As $x \to 0$: Substitute $x \to 0$ into the equation: $y^2 a = 0^3 \implies y^2 = 0.$ This indicates the curve passes through the origin.

• Asymptotic Behavior: The curve does not have any real linear asymptotes because its behavior does not approach a straight line in any direction.

Final Summary:

• Symmetry: The curve is symmetric about the x-axis.
• Asymptotes: There are no linear asymptotes.

Would you like me to explain any step in further detail? Let me know!

Related Questions:

1. What changes occur in the curve if $a < 0$ instead of $a > 0$?
2. How does the symmetry change if $y^2 (a - x) = -x^3$ instead?
3. Can you describe the behavior of the curve near $x = a$ in more detail?
4. What are the intercepts of the curve with the axes?
5. How would this curve differ in polar coordinates?

Tip:

Always test symmetry for $x$-axis, $y$-axis, and origin by substituting $-y$, $-x$, and $(-x, -y)$, respectively!