A ray of light along `x+sqrt(3)y=sqrt(3)` gets reflected upon reaching x-axis, the equation of the reflected ray is

Options

A. `y=x+sqrt(3)`
B. `sqrt(3)y =x-sqrt(3)`
C. `y =sqrt(3)x-sqrt(3)`
D. `sqrt(3)y= x-1`

Explanation

To solve the problem of finding the equation of the reflected ray of light after it reflects off the x-axis, we can follow these steps:

### Step 1: Write the equation of the incident ray
The equation of the incident ray is given as:
\[ x + \sqrt{3}y = \sqrt{3} \]

### Step 2: Find the point of intersection with the x-axis
To find where the ray intersects the x-axis, we set \( y = 0 \) in the equation of the incident ray:
\[ x + \sqrt{3}(0) = \sqrt{3} \implies x = \sqrt{3} \]
Thus, the point of intersection \( P \) is:
\[ P(\sqrt{3}, 0) \]

### Step 3: Determine the slope of the incident ray
We can rewrite the equation of the incident ray in slope-intercept form \( y = mx + c \):
\[ \sqrt{3}y = -x + \sqrt{3} \implies y = -\frac{1}{\sqrt{3}}x + \frac{1}{\sqrt{3}} \]
The slope \( m_1 \) of the incident ray is:
\[ m_1 = -\frac{1}{\sqrt{3}} \]

### Step 4: Find the angle of incidence
The angle of incidence \( \theta \) can be found using the tangent of the angle:
\[ \tan(\theta) = m_1 = -\frac{1}{\sqrt{3}} \]
This corresponds to an angle of \( 150^\circ \) (or \( \frac{5\pi}{6} \) radians).

### Step 5: Determine the angle of reflection
According to the law of reflection, the angle of reflection is equal to the angle of incidence. Therefore, the angle \( \theta' \) of the reflected ray with respect to the normal (which is vertical) is:
\[ \theta' = 90^\circ - \theta = 90^\circ - 60^\circ = 30^\circ \]

### Step 6: Find the slope of the reflected ray
The slope \( m_2 \) of the reflected ray can be calculated using:
\[ m_2 = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 7: Write the equation of the reflected ray
Using the point-slope form of the equation of a line, the equation of the reflected ray can be written as:
\[ y - y_1 = m_2(x - x_1) \]
Substituting \( P(\sqrt{3}, 0) \) and \( m_2 = \frac{1}{\sqrt{3}} \):
\[ y - 0 = \frac{1}{\sqrt{3}}(x - \sqrt{3}) \]
This simplifies to:
\[ \sqrt{3}y = x - \sqrt{3} \]
Rearranging gives us:
\[ x - \sqrt{3}y - \sqrt{3} = 0 \]

### Final Answer
Thus, the equation of the reflected ray is:
\[ x - \sqrt{3}y - \sqrt{3} = 0 \]

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