If the line `3x-2y=0` coincide with one of the lines given by `ax^(2)+2hxy+by^(2)=0`, then

Options

A. `4a+12h+9b=0`
B. `4a+12h-9b=0`
C. `4a-12h+9b=0`
D. `4a-12h-9b=0`

Answer

To solve the problem, we need to determine the conditions under which the line \(3x - 2y = 0\) coincides with one of the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\).

### Step-by-Step Solution:

1. **Identify the given line**: The line is given by the equation \(3x - 2y = 0\). We can rewrite this in slope-intercept form as \(y = \frac{3}{2}x\).

2. **Rewrite the conic equation**: The equation \(ax^2 + 2hxy + by^2 = 0\) represents a pair of straight lines. For this equation to represent two lines, the discriminant must be zero.

3. **Find the slopes of the lines represented by the conic**: The slopes of the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) can be found using the formula:
\[ m_1, m_2 = \frac{-2h \pm \sqrt{(2h)^2 - 4ab}}{2a} \]
Simplifying this gives:
\[ m_1, m_2 = \frac{-h \pm \sqrt{h^2 - ab}}{a} \]

4. **Set the slopes equal**: Since the line \(3x - 2y = 0\) has a slope of \(\frac{3}{2}\), we set this equal to the slopes obtained from the conic:
\[ \frac{-h + \sqrt{h^2 - ab}}{a} = \frac{3}{2} \quad \text{or} \quad \frac{-h - \sqrt{h^2 - ab}}{a} = \frac{3}{2} \]

5. **Solve for conditions**: We can solve these equations to find the conditions on \(a\), \(b\), and \(h\). Let's take the first equation:
\[ -h + \sqrt{h^2 - ab} = \frac{3a}{2} \]
Rearranging gives:
\[ \sqrt{h^2 - ab} = h + \frac{3a}{2} \]
Squaring both sides leads to:
\[ h^2 - ab = \left(h + \frac{3a}{2}\right)^2 \]
Expanding the right side:
\[ h^2 - ab = h^2 + 3ah + \frac{9a^2}{4} \]
This simplifies to:
\[ -ab = 3ah + \frac{9a^2}{4} \]
Rearranging gives:
\[ ab + 3ah + \frac{9a^2}{4} = 0 \]

6. **Conclusion**: The condition for the line \(3x - 2y = 0\) to coincide with one of the lines represented by the equation \(ax^2 + 2hxy + by^2 = 0\) is:
\[ ab + 3ah + \frac{9a^2}{4} = 0 \]