Sum of the possible values of `lamda` for which the following threes line
`x+y=1,lamdax+2y=3,lamda^(2)x+4y+9=0` are concurrent is

Options

A. -15
B. 14
C. 16
D. `-14`

Answer

To find the sum of the possible values of \( \lambda \) for which the lines \( x + y = 1 \), \( \lambda x + 2y = 3 \), and \( \lambda^2 x + 4y + 9 = 0 \) are concurrent, we will follow these steps:

### Step 1: Write the equations of the lines
The equations of the lines are:
1. \( L_1: x + y = 1 \)
2. \( L_2: \lambda x + 2y = 3 \)
3. \( L_3: \lambda^2 x + 4y + 9 = 0 \)

### Step 2: Find the intersection point of the first two lines
From the first line, we can express \( y \) in terms of \( x \):
\[ y = 1 - x \]

Substituting this into the second line:
\[ \lambda x + 2(1 - x) = 3 \]
Expanding this gives:
\[ \lambda x + 2 - 2x = 3 \]
Rearranging terms:
\[ (\lambda - 2)x + 2 = 3 \]
\[ (\lambda - 2)x = 1 \]
\[ x = \frac{1}{\lambda - 2} \]

Now substituting \( x \) back to find \( y \):
\[ y = 1 - \frac{1}{\lambda - 2} = \frac{\lambda - 3}{\lambda - 2} \]

Thus, the intersection point of the first two lines is:
\[ \left( \frac{1}{\lambda - 2}, \frac{\lambda - 3}{\lambda - 2} \right) \]

### Step 3: Substitute the intersection point into the third line
Now we need to check if this point satisfies the third line:
\[ \lambda^2 x + 4y + 9 = 0 \]
Substituting \( x \) and \( y \):
\[ \lambda^2 \left(\frac{1}{\lambda - 2}\right) + 4\left(\frac{\lambda - 3}{\lambda - 2}\right) + 9 = 0 \]
Multiplying through by \( \lambda - 2 \) to eliminate the denominator:
\[ \lambda^2 + 4(\lambda - 3) + 9(\lambda - 2) = 0 \]
Expanding this gives:
\[ \lambda^2 + 4\lambda - 12 + 9\lambda - 18 = 0 \]
Combining like terms:
\[ \lambda^2 + 13\lambda - 30 = 0 \]

### Step 4: Solve the quadratic equation
Now we can solve the quadratic equation:
\[ \lambda^2 + 13\lambda - 30 = 0 \]
Using the quadratic formula \( \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ \lambda = \frac{-13 \pm \sqrt{13^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ \lambda = \frac{-13 \pm \sqrt{169 + 120}}{2} \]
\[ \lambda = \frac{-13 \pm \sqrt{289}}{2} \]
\[ \lambda = \frac{-13 \pm 17}{2} \]
Calculating the two possible values:
1. \( \lambda = \frac{4}{2} = 2 \)
2. \( \lambda = \frac{-30}{2} = -15 \)

### Step 5: Check for valid values of \( \lambda \)
We need to check if \( \lambda = 2 \) is valid. If \( \lambda = 2 \), the denominator \( \lambda - 2 \) becomes zero, which is not valid since it leads to division by zero.

Thus, the only valid value is \( \lambda = -15 \).

### Step 6: Find the sum of the valid values
The sum of the possible values of \( \lambda \) is:
\[ -15 \]

### Final Answer
The sum of the possible values of \( \lambda \) for which the lines are concurrent is \( \boxed{-15} \).