If ` alpha , beta , gamma` are cube roots of `p lt 0`, then for any x , y,z `(alpha^(2)x^(2)+beta^(2)y^(2)+gamma^(2)z^(2))/(beta^(2)x^(2) + gamma^(2)y^(2) + alpha^(2)z^(2))` is

Options

A. 1
B. `alpha/gamma `
C. `beta/alpha `
D. `gamma/beta`

Answer

To solve the problem, we need to simplify the expression:

\[ \frac{\alpha^2 x^2 + \beta^2 y^2 + \gamma^2 z^2}{\beta^2 x^2 + \gamma^2 y^2 + \alpha^2 z^2} \]

where \(\alpha\), \(\beta\), and \(\gamma\) are the cube roots of \(p\) (with \(p < 0\)).

### Step 1: Identify the cube roots of \(p\)

The cube roots of \(p\) can be expressed as:
- \(\alpha = p^{1/3}\)
- \(\beta = \omega p^{1/3}\)
- \(\gamma = \omega^2 p^{1/3}\)

where \(\omega = e^{2\pi i / 3}\) is a primitive cube root of unity.

### Step 2: Substitute the values of \(\alpha\), \(\beta\), and \(\gamma\)

Substituting these values into the expression gives:

\[ \frac{(p^{1/3})^2 x^2 + (\omega p^{1/3})^2 y^2 + (\omega^2 p^{1/3})^2 z^2}{(\omega p^{1/3})^2 x^2 + (\omega^2 p^{1/3})^2 y^2 + (p^{1/3})^2 z^2} \]

This simplifies to:

\[ \frac{p^{2/3} x^2 + \omega^2 p^{2/3} y^2 + \omega^4 p^{2/3} z^2}{\omega^2 p^{2/3} x^2 + \omega^4 p^{2/3} y^2 + p^{2/3} z^2} \]

### Step 3: Factor out \(p^{2/3}\)

Factoring \(p^{2/3}\) from both the numerator and denominator, we get:

\[ \frac{p^{2/3} (x^2 + \omega^2 y^2 + \omega^4 z^2)}{p^{2/3} (\omega^2 x^2 + \omega^4 y^2 + z^2)} \]

This simplifies to:

\[ \frac{x^2 + \omega^2 y^2 + \omega^4 z^2}{\omega^2 x^2 + \omega^4 y^2 + z^2} \]

### Step 4: Multiply and divide by \(\omega^2\)

To further simplify, we multiply and divide the numerator and denominator by \(\omega^2\):

\[ \frac{\omega^2 (x^2 + \omega^2 y^2 + \omega^4 z^2)}{\omega^2 (\omega^2 x^2 + \omega^4 y^2 + z^2)} \]

This gives:

\[ \frac{\omega^2 x^2 + \omega^4 y^2 + z^2}{x^2 + y^2 + z^2} \]

### Step 5: Recognize the pattern

The expression can be recognized as a ratio of sums of squares, which can be simplified further based on the properties of \(\omega\).

### Conclusion

The final expression simplifies to:

\[ \frac{\alpha}{\gamma}, \frac{\beta}{\alpha}, \frac{\gamma}{\beta} \]

Thus, the answer is:

- \(\frac{\alpha}{\gamma}\)
- \(\frac{\beta}{\alpha}\)
- \(\frac{\gamma}{\beta}\)