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

Options

A. `1+sqrt(3)`
B. `2+sqrt(5)`
C. `1-sqrt(3)`
D. `2-sqrt(5)`

Answer

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

### Step-by-Step Solution:

1. **Identify Direction Ratios**:
Let the direction ratios of the line be:
\[ a = 1 + \lambda, \quad b = 1 - \lambda, \quad c = 2 \]

2. **Use the Cosine Formula**:
The cosine of the angle \( \theta \) between the line and the y-axis can be expressed as:
\[ \cos \theta = \frac{b}{\sqrt{a^2 + b^2 + c^2}} \]
Given that \( \theta = 60^\circ \), we have:
\[ \cos 60^\circ = \frac{1}{2} \]

3. **Set Up the Equation**:
Substitute \( b \) and \( \cos 60^\circ \) into the equation:
\[ \frac{1 - \lambda}{\sqrt{(1 + \lambda)^2 + (1 - \lambda)^2 + 2^2}} = \frac{1}{2} \]

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

5. **Expand and Simplify**:
Expand both sides:
\[ (1 - 2\lambda + \lambda^2) = \frac{1}{4} \left((1 + 2\lambda + \lambda^2) + (1 - 2\lambda + \lambda^2) + 4\right) \]
Simplifying the right side:
\[ = \frac{1}{4} \left(2 + 2\lambda^2 + 4\right) = \frac{1}{4} \left(6 + 2\lambda^2\right) = \frac{3}{2} + \frac{\lambda^2}{2} \]

6. **Rearranging the Equation**:
Now, equate both sides:
\[ 1 - 2\lambda + \lambda^2 = \frac{3}{2} + \frac{\lambda^2}{2} \]
Multiply through by 2 to eliminate the fraction:
\[ 2 - 4\lambda + 2\lambda^2 = 3 + \lambda^2 \]
Rearranging gives:
\[ 2\lambda^2 - \lambda^2 - 4\lambda + 2 - 3 = 0 \]
Simplifying leads to:
\[ \lambda^2 - 4\lambda - 1 = 0 \]

7. **Solve the Quadratic Equation**:
Using the quadratic formula:
\[ \lambda = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1, b = -4, c = -1 \):
\[ \lambda = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 + 4}}{2} = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2} = 2 \pm \sqrt{5} \]

8. **Final Values of \( \lambda \)**:
Thus, the possible values of \( \lambda \) are:
\[ \lambda = 2 + \sqrt{5} \quad \text{and} \quad \lambda = 2 - \sqrt{5} \]