If the direction ratios of a line are `1+lambda, 2-lambda, 4` and if it makes an angle of `60^(@)` with the y- axis, then the sum of the values of `lambda` is

Answer

To solve the problem, we need to find the sum of the values of \( \lambda \) given that the direction ratios of a line are \( 1 + \lambda, 2 - \lambda, 4 \) and that it makes an angle of \( 60^\circ \) with the y-axis.

### Step 1: Identify the direction ratios
The direction ratios of the line are given as:
- \( L_1 = 1 + \lambda \)
- \( M_1 = 2 - \lambda \)
- \( N_1 = 4 \)

The direction ratios of the y-axis are:
- \( L_2 = 0 \)
- \( M_2 = 1 \)
- \( N_2 = 0 \)

### Step 2: Use the formula for the cosine of the angle between two lines
The cosine of the angle \( \theta \) between two lines can be calculated using the formula:
\[ \cos \theta = \frac{L_1 L_2 + M_1 M_2 + N_1 N_2}{\sqrt{L_1^2 + M_1^2 + N_1^2} \sqrt{L_2^2 + M_2^2 + N_2^2}} \]
Substituting the values we have:
\[ \cos 60^\circ = \frac{(1 + \lambda)(0) + (2 - \lambda)(1) + (4)(0)}{\sqrt{(1 + \lambda)^2 + (2 - \lambda)^2 + 4^2} \sqrt{0^2 + 1^2 + 0^2}} \]
Since \( \cos 60^\circ = \frac{1}{2} \), we can simplify this to:
\[ \frac{2 - \lambda}{\sqrt{(1 + \lambda)^2 + (2 - \lambda)^2 + 16}} = \frac{1}{2} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 2(2 - \lambda) = \sqrt{(1 + \lambda)^2 + (2 - \lambda)^2 + 16} \]
Squaring both sides results in:
\[ 4(2 - \lambda)^2 = (1 + \lambda)^2 + (2 - \lambda)^2 + 16 \]

### Step 4: Expand both sides
Expanding the left-hand side:
\[ 4(4 - 4\lambda + \lambda^2) = 16 - 16\lambda + 4\lambda^2 \]
Expanding the right-hand side:
\[ (1 + 2\lambda + \lambda^2) + (4 - 4\lambda + \lambda^2) + 16 = 2\lambda^2 - 2\lambda + 21 \]

### Step 5: Set the equation to zero
Now we have:
\[ 16 - 16\lambda + 4\lambda^2 = 2\lambda^2 - 2\lambda + 21 \]
Rearranging gives:
\[ 4\lambda^2 - 2\lambda + 16 - 21 + 2\lambda = 0 \]
This simplifies to:
\[ 2\lambda^2 - 14 = 0 \]

### Step 6: Solve for \( \lambda \)
Dividing through by 2:
\[ \lambda^2 - 7 = 0 \]
Factoring gives:
\[ (\lambda - \sqrt{7})(\lambda + \sqrt{7}) = 0 \]
Thus, the solutions for \( \lambda \) are:
\[ \lambda = \sqrt{7} \quad \text{and} \quad \lambda = -\sqrt{7} \]

### Step 7: Find the sum of the values of \( \lambda \)
The sum of the values of \( \lambda \) is:
\[ \sqrt{7} + (-\sqrt{7}) = 0 \]

### Final Answer
The sum of the values of \( \lambda \) is \( 0 \).