The larger of two angles made with the X-axis of a straight line drawn through (1, 2) so that it intersects the line x + y = 4 at a paint distant `sqrt(6)` / 3 from the point (1, 2) is

Options

A. `60^(@)`
B. `75^(@)`
C. `105^(@)`
D. none of these

Answer

To solve the problem, we need to find the larger of the two angles made with the X-axis by a straight line that passes through the point (1, 2) and 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 line equation**: The line \( x + y = 4 \) can be rewritten in slope-intercept form as \( y = -x + 4 \). The slope of this line is -1.

2. **Determine the point of intersection**: Let the point of intersection of the line through (1, 2) and the line \( x + y = 4 \) be denoted as \( P(x, y) \). Since \( P \) lies on the line \( x + y = 4 \), we have:
\[ y = 4 - x \]

3. **Calculate the distance from (1, 2) to P**: The distance \( d \) from the point (1, 2) to the point \( P(x, y) \) is given by the distance formula:
\[ d = \sqrt{(x - 1)^2 + (y - 2)^2} \]
Substituting \( y = 4 - x \) into the distance formula gives:
\[ d = \sqrt{(x - 1)^2 + ((4 - x) - 2)^2} = \sqrt{(x - 1)^2 + (2 - x)^2} \]

4. **Set the distance equal to \( \frac{\sqrt{6}}{3} \)**: We know that the distance \( d \) is \( \frac{\sqrt{6}}{3} \), thus:
\[ \sqrt{(x - 1)^2 + (2 - x)^2} = \frac{\sqrt{6}}{3} \]
Squaring both sides gives:
\[ (x - 1)^2 + (2 - x)^2 = \frac{6}{9} = \frac{2}{3} \]

5. **Simplify the equation**: Expanding the left side:
\[ (x - 1)^2 = x^2 - 2x + 1 \]
\[ (2 - x)^2 = 4 - 4x + x^2 \]
Combining these:
\[ x^2 - 2x + 1 + 4 - 4x + x^2 = \frac{2}{3} \]
\[ 2x^2 - 6x + 5 = \frac{2}{3} \]
Multiplying through by 3 to eliminate the fraction:
\[ 6x^2 - 18x + 15 = 2 \]
\[ 6x^2 - 18x + 13 = 0 \]

6. **Solve the quadratic equation**: Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 6 \cdot 13}}{2 \cdot 6} \]
\[ x = \frac{18 \pm \sqrt{324 - 312}}{12} \]
\[ x = \frac{18 \pm \sqrt{12}}{12} = \frac{18 \pm 2\sqrt{3}}{12} = \frac{3 \pm \frac{\sqrt{3}}{3}}{2} \]

7. **Find the corresponding y-values**: Substitute \( x \) back into \( y = 4 - x \) to find the y-coordinates.

8. **Calculate the angles**: The slope \( m \) of the line through (1, 2) and \( P(x, y) \) is given by:
\[ m = \frac{y - 2}{x - 1} \]
The angle \( \theta \) with respect to the x-axis can be found using:
\[ \tan(\theta) = m \]
Calculate the angles \( \theta_1 \) and \( \theta_2 \) and identify the larger angle.