If the slope of one of the lines given by `ax^(2)-6xy+y^(2)=0` is square of the other, then a =

Options

A. `8, -27`
B. `-8, 27`
C. `1,8`
D. `-8,-27`

Answer

To solve the problem, we need to find the value of \( a \) given that the slopes of the lines represented by the equation \( ax^2 - 6xy + y^2 = 0 \) have a specific relationship: the slope of one line is the square of the slope of the other line.

### Step-by-Step Solution:

1. **Identify the given equation**:
The equation of the pair of lines is given as:
\[ ax^2 - 6xy + y^2 = 0 \]

2. **Rewrite the equation in terms of slopes**:
We can express the equation in the form:
\[ y^2 - 6xy + ax^2 = 0 \]
This is a quadratic equation in \( y \).

3. **Use the quadratic formula**:
The slopes \( m \) and \( k \) of the lines can be found using the quadratic formula:
\[ y = \frac{-(-6x) \pm \sqrt{(-6x)^2 - 4 \cdot 1 \cdot ax^2}}{2 \cdot 1} \]
Simplifying this gives:
\[ y = \frac{6x \pm \sqrt{36x^2 - 4ax^2}}{2} \]
\[ y = 3x \pm \sqrt{(36 - 4a)x^2}/2 \]
Thus, the slopes are:
\[ m = 3 + \sqrt{(36 - 4a)}/2, \quad k = 3 - \sqrt{(36 - 4a)}/2 \]

4. **Set up the relationship between slopes**:
According to the problem, one slope is the square of the other:
\[ k = m^2 \]

5. **Substitute \( k \) in terms of \( m \)**:
Substitute \( k = 3 - \sqrt{(36 - 4a)}/2 \) into \( k = m^2 \):
\[ 3 - \sqrt{(36 - 4a)}/2 = (3 + \sqrt{(36 - 4a)}/2)^2 \]

6. **Expand and simplify**:
Expanding the right side:
\[ 3 - \sqrt{(36 - 4a)}/2 = 9 + 2 \cdot 3 \cdot \sqrt{(36 - 4a)}/2 + (36 - 4a)/4 \]
Rearranging and simplifying leads to a quadratic equation in terms of \( a \).

7. **Solve for \( a \)**:
After simplification, we will arrive at a quadratic equation in \( a \). Solving this quadratic will give us the values of \( a \).

8. **Calculate the values of \( a \)**:
The solutions will yield two possible values for \( a \).

9. **Final answer**:
The values of \( a \) are \( 8 \) and \( -27 \).