Key Concepts and Formulas

• Law of Reflection: The angle of incidence equals the angle of reflection. When reflecting off the x-axis, the angle the incident ray makes with the x-axis is the same as the angle the reflected ray makes with the x-axis.
• Slope-intercept form: The equation of a line is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.
• Point-slope form: The equation of a line passing through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$.

Step-by-Step Solution

Step 1: Analyze the Incident Ray and Convert to Slope-Intercept Form

The equation of the incident ray is given as $x + \sqrt{3}y = \sqrt{3}$. To easily identify the slope, we rewrite this equation in slope-intercept form ($y = mx + c$). $x + \sqrt{3}y = \sqrt{3}$ $\sqrt{3}y = -x + \sqrt{3}$ $y = -\frac{1}{\sqrt{3}}x + 1$ The slope of the incident ray is $m_1 = -\frac{1}{\sqrt{3}}$.

Step 2: Determine the Angle of Incidence

The slope of a line is related to the angle it makes with the x-axis by $m = \tan \theta$. In this case, $\tan \theta_1 = -\frac{1}{\sqrt{3}}$. Since we want the acute angle the incident ray makes with the x-axis, we take the absolute value: $\tan \alpha = \left|-\frac{1}{\sqrt{3}}\right| = \frac{1}{\sqrt{3}}$ $\alpha = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \text{ or } 30^\circ$ This is the angle of incidence.

Step 3: Find the Point of Incidence

The point of incidence is where the ray hits the x-axis, meaning $y = 0$. Substitute $y = 0$ into the equation of the incident ray: $x + \sqrt{3}(0) = \sqrt{3}$ $x = \sqrt{3}$ So, the point of incidence is $B = (\sqrt{3}, 0)$.

Step 4: Determine the Slope of the Reflected Ray

Since the angle of incidence equals the angle of reflection, the reflected ray also makes an angle of $\alpha = \frac{\pi}{6}$ with the x-axis. Because the incident ray has a negative slope, the reflected ray will have a positive slope. Therefore, the slope of the reflected ray is: $m_2 = \tan \alpha = \tan \left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

Step 5: Find the Equation of the Reflected Ray

Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with the point of incidence $B(\sqrt{3}, 0)$ and the slope $m_2 = \frac{1}{\sqrt{3}}$, we get: $y - 0 = \frac{1}{\sqrt{3}}(x - \sqrt{3})$ $y = \frac{1}{\sqrt{3}}x - \frac{\sqrt{3}}{\sqrt{3}}$ $y = \frac{1}{\sqrt{3}}x - 1$ Multiply by $\sqrt{3}$ to eliminate the fraction: $\sqrt{3}y = x - \sqrt{3}$

Step 6: Transform to Required Form We need to rearrange this into the forms provided in the options. Option A is $y = x + \sqrt 3$, which we know is wrong since it has the wrong slope. Option B is $\sqrt{3}y = x - \sqrt 3$, which matches our result. Therefore, we can rearrange option A to match.

Common Mistakes & Tips

• Sign of the Slope: Remember to change the sign of the slope when reflecting across the x-axis. A negative slope becomes positive, and vice versa.
• Acute Angle: Always use the acute angle that the line makes with the x-axis when applying the law of reflection.
• Point-Slope Form: The point-slope form is a convenient way to find the equation of a line when you know a point and the slope.

Summary

We found the equation of the reflected ray by first determining the slope and point of incidence of the incident ray. We then used the law of reflection to find the slope of the reflected ray and, together with the point of incidence, formulated the equation of the reflected ray in point-slope form. Finally, we simplified the equation to match one of the given options. The equation of the reflected ray is $\sqrt{3}y = x - \sqrt{3}$.

Final Answer

The final answer is \boxed{\sqrt{3}y = x - \sqrt{3}}, which corresponds to option (B).