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 values of \( \lambda \) given the direction ratios of a line and the angle it makes with the y-axis.

### Step-by-Step Solution:

1. **Identify the Direction Ratios**:
The direction ratios of the line are given as:
\[ l = 1 + \lambda, \quad m = 2 - \lambda, \quad n = 4 \]

2. **Use the Angle with the y-axis**:
The line makes an angle of \( 60^\circ \) with the y-axis. The cosine of the angle \( \beta \) with the y-axis is:
\[ \cos \beta = \cos 60^\circ = \frac{1}{2} \]

3. **Relate Cosine to Direction Ratios**:
The cosine of the angle \( \beta \) can also be expressed using the direction ratios:
\[ \cos \beta = \frac{m}{\sqrt{l^2 + m^2 + n^2}} \]
Substituting the values:
\[ \frac{2 - \lambda}{\sqrt{(1 + \lambda)^2 + (2 - \lambda)^2 + 4^2}} = \frac{1}{2} \]

4. **Square Both Sides**:
Squaring both sides gives:
\[ \left(2 - \lambda\right)^2 = \frac{1}{4} \left((1 + \lambda)^2 + (2 - \lambda)^2 + 16\right) \]

5. **Expand Both Sides**:
Expanding the left side:
\[ (2 - \lambda)^2 = 4 - 4\lambda + \lambda^2 \]
Expanding the right side:
\[ (1 + \lambda)^2 = 1 + 2\lambda + \lambda^2 \]
\[ (2 - \lambda)^2 = 4 - 4\lambda + \lambda^2 \]
Thus,
\[ 1 + 2\lambda + \lambda^2 + 4 - 4\lambda + \lambda^2 + 16 = 21 - 2\lambda + 2\lambda^2 \]
Therefore, the equation becomes:
\[ 4 - 4\lambda + \lambda^2 = \frac{1}{4} (21 - 2\lambda + 2\lambda^2) \]

6. **Multiply Through by 4**:
To eliminate the fraction:
\[ 16 - 16\lambda + 4\lambda^2 = 21 - 2\lambda + 2\lambda^2 \]

7. **Rearrange the Equation**:
Bringing all terms to one side:
\[ 4\lambda^2 - 2\lambda - 5 = 0 \]

8. **Use the Quadratic Formula**:
The quadratic formula is given by:
\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 4, b = -2, c = -5 \):
\[ \lambda = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 4 \cdot (-5)}}{2 \cdot 4} \]
\[ = \frac{2 \pm \sqrt{4 + 80}}{8} \]
\[ = \frac{2 \pm \sqrt{84}}{8} \]
\[ = \frac{2 \pm 2\sqrt{21}}{8} = \frac{1 \pm \sqrt{21}}{4} \]

9. **Sum of the Values of \( \lambda \)**:
The two values of \( \lambda \) are:
\[ \lambda_1 = \frac{1 + \sqrt{21}}{4}, \quad \lambda_2 = \frac{1 - \sqrt{21}}{4} \]
The sum of the values of \( \lambda \) is:
\[ \lambda_1 + \lambda_2 = \frac{1 + \sqrt{21}}{4} + \frac{1 - \sqrt{21}}{4} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer:
The sum of the values of \( \lambda \) is:
\[ \frac{1}{2} \]