Key Concepts and Formulas

• Slope and Angle: The slope $m$ of a line is related to the angle $\theta$ it makes with the positive x-axis by $m = \tan \theta$.
• Equation of a Line (Point-Slope Form): The equation of a line passing through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$.
• Perpendicular Distance: The perpendicular distance from $(x_0, y_0)$ to $Ax + By + C = 0$ is $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Determine the new angle and relate it to the given slope.

• We are given that the original line makes an angle $\alpha$ with the positive x-axis. It's rotated clockwise by $\frac{\alpha}{2}$.
• This means the new angle, $\theta_{new}$, is given by $\theta_{new} = \alpha - \frac{\alpha}{2} = \frac{\alpha}{2}$.
• The slope of the new line is given as $2 - \sqrt{3}$. Using the slope-angle relationship: $\tan \left(\frac{\alpha}{2}\right) = 2 - \sqrt{3}$

Step 2: Find the value of $\alpha$.

• We recognize that $2 - \sqrt{3}$ is a standard tangent value. Recall that $\tan 15^\circ = 2 - \sqrt{3}$.
• Therefore, $\frac{\alpha}{2} = 15^\circ$, which implies $\alpha = 30^\circ$.

Step 3: Determine the equation of the rotated line.

• The rotated line passes through the point $P(a, 0)$ and has a slope of $2 - \sqrt{3}$.
• Using the point-slope form, the equation of the rotated line is: $y - 0 = (2 - \sqrt{3})(x - a)$ $y = (2 - \sqrt{3})x - (2 - \sqrt{3})a$
• Rearranging the equation into the general form $Ax + By + C = 0$: $(2 - \sqrt{3})x - y - (2 - \sqrt{3})a = 0$

Step 4: Use the distance from the origin to find the value of $a$.

• The distance of the rotated line from the origin is given as $\frac{1}{\sqrt{2}}$.
• Using the perpendicular distance formula: $\frac{|(2 - \sqrt{3})(0) - (0) - (2 - \sqrt{3})a|}{\sqrt{(2 - \sqrt{3})^2 + (-1)^2}} = \frac{1}{\sqrt{2}}$ $\frac{|-(2 - \sqrt{3})a|}{\sqrt{(4 + 3 - 4\sqrt{3}) + 1}} = \frac{1}{\sqrt{2}}$ $\frac{|(2 - \sqrt{3})a|}{\sqrt{8 - 4\sqrt{3}}} = \frac{1}{\sqrt{2}}$ $|(2 - \sqrt{3})a| = \frac{\sqrt{8 - 4\sqrt{3}}}{\sqrt{2}}$ $|(2 - \sqrt{3})a| = \sqrt{\frac{8 - 4\sqrt{3}}{2}} = \sqrt{4 - 2\sqrt{3}}$ $|(2 - \sqrt{3})a| = \sqrt{3 - 2\sqrt{3} + 1} = \sqrt{(\sqrt{3} - 1)^2} = |\sqrt{3} - 1| = \sqrt{3} - 1$
• Therefore, $|(2 - \sqrt{3})a| = \sqrt{3} - 1$, which means $|a| = \frac{\sqrt{3} - 1}{2 - \sqrt{3}}$.
• Rationalize the denominator: $|a| = \frac{(\sqrt{3} - 1)(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} = \frac{2\sqrt{3} + 3 - 2 - \sqrt{3}}{4 - 3} = \frac{\sqrt{3} + 1}{1} = \sqrt{3} + 1$.
• So, $a = \pm (\sqrt{3} + 1)$.

Step 5: Calculate the value of the expression $3a^2 \tan^2 \alpha - 2\sqrt{3}$.

• We have $\alpha = 30^\circ$, so $\tan \alpha = \tan 30^\circ = \frac{1}{\sqrt{3}}$.
• We also have $a^2 = (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}$.
• Substituting these values into the expression: $3a^2 \tan^2 \alpha - 2\sqrt{3} = 3(4 + 2\sqrt{3})\left(\frac{1}{\sqrt{3}}\right)^2 - 2\sqrt{3}$ $= 3(4 + 2\sqrt{3})\left(\frac{1}{3}\right) - 2\sqrt{3} = (4 + 2\sqrt{3}) - 2\sqrt{3} = 4$

Common Mistakes & Tips

• Trigonometric Values: Remember the standard trigonometric values, especially for angles like 15°, 30°, 45°, 60°, and 90°.
• Rationalization: Rationalizing the denominator is a common technique for simplifying expressions with surds.
• Careful with Signs: Double-check your signs when applying the distance formula and rearranging equations.

Summary

We determined the new angle after rotation and found $\alpha = 30^\circ$. Using the perpendicular distance from the origin, we calculated $a^2 = (\sqrt{3} + 1)^2$. Finally, we substituted these values into the expression $3a^2 \tan^2 \alpha - 2\sqrt{3}$ to obtain the result 4.

The final answer is \boxed{4}, which corresponds to option (B).