If the equation `4y^(3) - 8a^(2)yx^(2) - 3ay^(2)x +8x^(3) = 0` represents three straight lines, two of them are perpendicular, then sum of all possible values of a is equal to

Options

A. `(3)/(8)`
B. `(-3)/(4)`
C. `(1)/(4)`
D. `-2`

Explanation

To solve the equation \(4y^3 - 8a^2yx^2 - 3ay^2x + 8x^3 = 0\) representing three straight lines, where two of them are perpendicular, we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:

\[ 4y^3 - 8a^2yx^2 - 3ay^2x + 8x^3 = 0 \]

We can factor out \(x^3\) from the equation:

\[ x^3 \left(4 \frac{y^3}{x^3} - 8a^2 \frac{y}{x^2} - 3a \frac{y^2}{x} + 8\right) = 0 \]

This simplifies to:

\[ 4\left(\frac{y}{x}\right)^3 - 8a^2\left(\frac{y}{x}\right) - 3a\left(\frac{y}{x}\right)^2 + 8 = 0 \]

Let \(t = \frac{y}{x}\). The equation becomes:

\[ 4t^3 - 3at^2 - 8a^2t + 8 = 0 \]

### Step 2: Identifying the Roots
We denote the roots of this cubic equation as \(m_1, m_2, m_3\). According to Vieta's formulas, the product of the roots is given by:

\[ m_1 m_2 m_3 = -\frac{\text{constant term}}{\text{leading coefficient}} = -\frac{8}{4} = -2 \]

### Step 3: Condition for Perpendicular Lines
Since two lines are perpendicular, we can assume \(m_1 m_2 = -1\). Thus, we have:

\[ m_1 m_2 m_3 = -2 \implies (-1)m_3 = -2 \implies m_3 = 2 \]

### Step 4: Using the Roots to Form the Equation
Now we substitute \(m_3 = 2\) back into the polynomial. The roots can be expressed as \(m_1, m_2, 2\). The sum of the roots is:

\[ m_1 + m_2 + 2 = \frac{3a}{4} \]

### Step 5: Finding the Relationship
From the product of the roots, we have:

\[ m_1 m_2 \cdot 2 = -2 \implies m_1 m_2 = -1 \]

### Step 6: Forming the Quadratic Equation
Now we can express \(m_1\) and \(m_2\) as the roots of the quadratic equation:

\[ t^2 - \left(\frac{3a}{4} - 2\right)t - 1 = 0 \]

### Step 7: Solving for \(a\)
The quadratic equation can be simplified and solved for \(a\):

1. The sum of the roots gives us the equation \(m_1 + m_2 = \frac{3a}{4} - 2\).
2. The product of the roots gives us \(m_1 m_2 = -1\).

Using the quadratic formula, we can find the values of \(a\):

\[ \text{Sum of roots} = -\frac{B}{A} = -\frac{3}{4} \]

### Final Calculation
Thus, the sum of all possible values of \(a\) is:

\[ \text{Sum of all possible values of } a = -\frac{3}{4} \]

### Conclusion
The final answer is:

\[ \text{Sum of all possible values of } a = -\frac{3}{4} \]