Angle made with the x-axis by a straight line drawn through (1, 2) so that it intersects `x+y=4` at a distance `(sqrt(6))/3` from (1, 2) is (a)`105^0` (b) `75^0` (c) `60^0` (d) `15^0`

Options

A. `105^(@)`
B. `75^(@)`
C. `60^(@)`
D. `15^(@)`

Explanation

To solve the problem, we need to find the angle made with the x-axis by a straight line drawn through the point (1, 2) that intersects the line \(x + y = 4\) at a distance of \(\frac{\sqrt{6}}{3}\) from the point (1, 2).

### Step-by-Step Solution:

1. **Identify the given information**:
- Point through which the line passes: \( (1, 2) \)
- Distance from the point to the intersection: \( r = \frac{\sqrt{6}}{3} \)
- Line equation: \( x + y = 4 \)

2. **Use the parametric form of the line**:
The parametric form of a line passing through a point \((a, b)\) at a distance \(r\) is given by:
\[ \frac{x - a}{\cos \theta} = \frac{y - b}{\sin \theta} = r \]
Substituting \(a = 1\), \(b = 2\), and \(r = \frac{\sqrt{6}}{3}\):
\[ \frac{x - 1}{\cos \theta} = \frac{y - 2}{\sin \theta} = \frac{\sqrt{6}}{3} \]

3. **Express \(x\) and \(y\) in terms of \(\theta\)**:
From the equations above:
\[ x - 1 = \frac{\sqrt{6}}{3} \cos \theta \implies x = \frac{\sqrt{6}}{3} \cos \theta + 1 \]
\[ y - 2 = \frac{\sqrt{6}}{3} \sin \theta \implies y = \frac{\sqrt{6}}{3} \sin \theta + 2 \]

4. **Substitute \(x\) and \(y\) into the line equation**:
Substitute \(x\) and \(y\) into the line equation \(x + y = 4\):
\[ \left(\frac{\sqrt{6}}{3} \cos \theta + 1\right) + \left(\frac{\sqrt{6}}{3} \sin \theta + 2\right) = 4 \]
Simplifying this gives:
\[ \frac{\sqrt{6}}{3} (\cos \theta + \sin \theta) + 3 = 4 \]
\[ \frac{\sqrt{6}}{3} (\cos \theta + \sin \theta) = 1 \]
\[ \cos \theta + \sin \theta = \frac{3}{\sqrt{6}} = \frac{\sqrt{6}}{2} \]

5. **Square both sides**:
Squaring both sides:
\[ (\cos \theta + \sin \theta)^2 = \left(\frac{\sqrt{6}}{2}\right)^2 \]
Expanding the left side:
\[ \cos^2 \theta + 2 \cos \theta \sin \theta + \sin^2 \theta = \frac{6}{4} = \frac{3}{2} \]
Since \(\cos^2 \theta + \sin^2 \theta = 1\):
\[ 1 + 2 \cos \theta \sin \theta = \frac{3}{2} \]
\[ 2 \cos \theta \sin \theta = \frac{3}{2} - 1 = \frac{1}{2} \]
\[ \cos \theta \sin \theta = \frac{1}{4} \]

6. **Use the identity for sine**:
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ \sin 2\theta = 2 \cdot \frac{1}{4} = \frac{1}{2} \]
Therefore, \(2\theta = 30^\circ\) or \(2\theta = 150^\circ\).

7. **Find \(\theta\)**:
Thus, \(\theta = 15^\circ\) or \(\theta = 75^\circ\).

8. **Final answer**:
The angles made with the x-axis by the line are \(15^\circ\) and \(75^\circ\).

### Conclusion:
The correct options are:
- (b) \(75^\circ\)
- (d) \(15^\circ\)