Trace the curve 3ay^2 = x^2(a-x) with diagram...........

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To trace the curve defined by the equation 3ay² = x²(a - x), we first need to understand its components and then analyze it step by step. This equation is a type of quadratic curve, and we can derive its shape through a series of transformations and evaluations.

Understanding the Equation

The equation 3ay² = x²(a - x) can be rewritten for clarity:

• Rearranging: y² = (x²(a - x))/(3a)
• Identifying Variables: Here, 'a' is a constant that affects the shape and position of the curve.

Finding Key Points

To effectively trace the curve, we should find key points. Let's start by analyzing some specific values of 'x' to find corresponding 'y' values:

• At x = 0:
  • y² = 0, thus y = 0.
• At x = a:
  • y² = 0, thus y = 0.
• At x = a/2:
  • y² = (a²/4)(a/2)/(3a) = a²/24, thus y = ±√(a²/24) = ±(a/2√6).

Symmetry and Behavior

This curve is symmetric with respect to the x-axis because the equation involves y², meaning for every positive value of y, there is a corresponding negative value. This symmetry simplifies our sketching process.

Sketching the Curve

Now, let’s sketch the curve based on the points we’ve calculated. The key points we have are:

• (0, 0)
• (a, 0)
• (a/2, a/2√6) and (a/2, -a/2√6)

1. **Plot the Points**: Start by marking the points on a Cartesian plane.
2. **Draw the Axes**: Label your axes with appropriate scales for both x and y.
3. **Connect the Points**: Since the curve is symmetrical, you will notice it bulges upward and downward around the central point (a/2, ±a/2√6).

Visual Representation

While I can't draw here, imagine the curve resembling a parabolic shape that opens horizontally. The x-intercepts at (0,0) and (a,0) indicate where the curve meets the x-axis, while the points (a/2, ±a/2√6) indicate the maximum height of the curve above and below the x-axis.

Conclusion of Analysis

Through this step-by-step interpretation, you can see how the equation describes a curve that is symmetric and shaped similarly to a parabola. The parameter 'a' influences the width and height of the curve, allowing for variations based on its value. This analysis equips you to approach similar problems with confidence.