If `alpha, beta gamma` are the real roots of the equation `x^(3)-3px^(2)+3qx-1=0`, then find the centroid of the triangle whose vertices are `(alpha, (1)/(alpha)), (beta, (1)/(beta))` and `(gamma, (1)/(gamma))`.

Answer

To find the centroid of the triangle whose vertices are \((\alpha, \frac{1}{\alpha}), (\beta, \frac{1}{\beta}), (\gamma, \frac{1}{\gamma})\), where \(\alpha, \beta, \gamma\) are the roots of the polynomial equation \(x^3 - 3px^2 + 3qx - 1 = 0\), we can follow these steps:

### Step 1: Identify the sum of the roots
From Vieta's formulas, we know:
- The sum of the roots \(\alpha + \beta + \gamma = 3p\).

### Step 2: Identify the sum of the products of the roots taken two at a time
Again from Vieta's formulas:
- The sum of the products of the roots taken two at a time \(\alpha\beta + \beta\gamma + \gamma\alpha = 3q\).

### Step 3: Identify the product of the roots
From Vieta's formulas:
- The product of the roots \(\alpha\beta\gamma = 1\).

### Step 4: Calculate the coordinates of the centroid
The formula for the centroid \(G\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[ G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
In our case, the vertices are:
- \(A = (\alpha, \frac{1}{\alpha})\)
- \(B = (\beta, \frac{1}{\beta})\)
- \(C = (\gamma, \frac{1}{\gamma})\)

Thus, we can calculate the coordinates of the centroid as follows:

1. **Calculate the x-coordinate of the centroid**:
\[ x_G = \frac{\alpha + \beta + \gamma}{3} = \frac{3p}{3} = p \]

2. **Calculate the y-coordinate of the centroid**:
\[ y_G = \frac{\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}}{3} \]
We can simplify \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\) using the identity:
\[ \frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} = \frac{\beta\gamma + \gamma\alpha + \alpha\beta}{\alpha\beta\gamma} = \frac{3q}{1} = 3q \]
Therefore,
\[ y_G = \frac{3q}{3} = q \]

### Step 5: Write the final coordinates of the centroid
Thus, the coordinates of the centroid \(G\) are:
\[ G = (p, q) \]

### Final Answer
The centroid of the triangle is \((p, q)\).

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