If `alpha, beta, gamma` be the roots of `x^(3) + a^(3) = 0 (a in R)`, then the number of equation(s) whose roots are `((alpha)/(beta))^(2) and ((alpha)/(gamma))^(2)`, is

Options

A. 1
B. 2
C. 3
D. 6

Answer

To solve the problem, we need to find the number of equations whose roots are \(\left(\frac{\alpha}{\beta}\right)^2\) and \(\left(\frac{\alpha}{\gamma}\right)^2\), given that \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3 + a^3 = 0\).

### Step 1: Identify the roots of the given equation
The equation \(x^3 + a^3 = 0\) can be rewritten as:
\[ x^3 = -a^3 \]
The roots of this equation can be expressed as:
\[ \alpha = -a, \quad \beta = -a\omega, \quad \gamma = -a\omega^2 \]
where \(\omega\) is a primitive cube root of unity, defined as \(\omega = e^{2\pi i / 3}\). The roots are:
\[ \alpha = -a, \quad \beta = -a\omega, \quad \gamma = -a\omega^2 \]

### Step 2: Calculate \(\left(\frac{\alpha}{\beta}\right)^2\) and \(\left(\frac{\alpha}{\gamma}\right)^2\)
Now we calculate the values:
\[ \frac{\alpha}{\beta} = \frac{-a}{-a\omega} = \frac{1}{\omega} \quad \Rightarrow \quad \left(\frac{\alpha}{\beta}\right)^2 = \left(\frac{1}{\omega}\right)^2 = \frac{1}{\omega^2} \]
Similarly,
\[ \frac{\alpha}{\gamma} = \frac{-a}{-a\omega^2} = \frac{1}{\omega^2} \quad \Rightarrow \quad \left(\frac{\alpha}{\gamma}\right)^2 = \left(\frac{1}{\omega^2}\right)^2 = \frac{1}{\omega^4} = \omega \]

### Step 3: Find the sum and product of the roots
Now we have the roots:
\[ r_1 = \frac{1}{\omega^2}, \quad r_2 = \omega \]
Next, we calculate the sum \(S\) and product \(P\) of the roots:
\[ S = r_1 + r_2 = \frac{1}{\omega^2} + \omega \]
Using the property of cube roots of unity, we know:
\[ 1 + \omega + \omega^2 = 0 \quad \Rightarrow \quad \omega + \frac{1}{\omega^2} = -1 \]
Thus,
\[ S = -1 \]

For the product:
\[ P = r_1 \cdot r_2 = \frac{1}{\omega^2} \cdot \omega = \frac{\omega}{\omega^2} = \frac{1}{\omega} = \omega^2 \]

### Step 4: Form the quadratic equation
Using the sum and product of the roots, we can form the quadratic equation:
\[ x^2 - Sx + P = 0 \quad \Rightarrow \quad x^2 + x + \omega^2 = 0 \]

### Step 5: Determine the number of equations
Since the coefficients of the quadratic equation depend on the roots, and we have determined the sum and product uniquely, there is only one quadratic equation that can be formed with these roots.

### Final Answer
The number of equations whose roots are \(\left(\frac{\alpha}{\beta}\right)^2\) and \(\left(\frac{\alpha}{\gamma}\right)^2\) is:
\[ \boxed{1} \]