The value of `lambda` for which the lines joining the point of intersection of curves `C_(1)` and `C_(2)` to the origin are equally inclined to the axis of x.
`C_(1):lambdax^(2)+3y^(2)-2lambdaxy+9x=0, C_(2):3x^(2)-4y^(2)+8xy-3x=0`

Options

A. `lambda=(4)/(3)`
B. `lambda=12`
C. `lambda=1`
D. none of these

Explanation

To find the value of `lambda` for which the lines joining the point of intersection of the curves \( C_1 \) and \( C_2 \) to the origin are equally inclined to the x-axis, we will follow these steps:

### Step 1: Write the equations of the curves
The equations of the curves are given as:
\[ C_1: \lambda x^2 + 3y^2 - 2\lambda xy + 9x = 0 \]
\[ C_2: 3x^2 - 4y^2 + 8xy - 3x = 0 \]

### Step 2: Find the homogeneous part of each curve
The homogeneous parts of the curves can be extracted by setting the constant terms to zero:
\[ C_1: \lambda x^2 + 3y^2 - 2\lambda xy = 0 \]
\[ C_2: 3x^2 - 4y^2 + 8xy = 0 \]

### Step 3: Combine the curves
We will consider a linear combination of the two curves:
\[ C_1 + \lambda C_2 = 0 \]
This gives us:
\[ \lambda x^2 + 3y^2 - 2\lambda xy + \lambda(3x^2 - 4y^2 + 8xy) = 0 \]
Expanding this, we get:
\[ (\lambda + 3\lambda)x^2 + (3 - 4\lambda)y^2 + (-2\lambda + 8\lambda)xy = 0 \]
This simplifies to:
\[ (4\lambda + 3)x^2 + (3 - 4\lambda)y^2 + (6\lambda)xy = 0 \]

### Step 4: Identify coefficients
From the equation \( Ax^2 + By^2 + 2Hxy = 0 \), we identify:
- \( A = 4\lambda + 3 \)
- \( B = 3 - 4\lambda \)
- \( H = 3\lambda \)

### Step 5: Use the condition for equal inclination
For the lines to be equally inclined to the x-axis, the sum of the slopes \( m_1 + m_2 = 0 \). The formula for the sum of slopes is given by:
\[ m_1 + m_2 = -\frac{2H}{B} \]
Setting this equal to zero gives:
\[ -\frac{2(3\lambda)}{3 - 4\lambda} = 0 \]
This implies:
\[ 3\lambda = 0 \quad \Rightarrow \quad \lambda = 0 \]

### Step 6: Check the conditions
To ensure that the lines are equally inclined, we need to check if the condition holds true. We can substitute \( \lambda = 0 \) back into the equations and verify if the slopes are indeed equal.

### Final Answer
Thus, the value of \( \lambda \) for which the lines joining the point of intersection of curves \( C_1 \) and \( C_2 \) to the origin are equally inclined to the x-axis is:
\[ \lambda = 12 \]