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`

Answer

To solve the problem of finding the equation of the reflected ray of light, we will follow these steps:

### Step 1: Identify the given line equation
The equation of the incoming ray of light is given as:
\[ x + \sqrt{3}y = \sqrt{3} \]

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

### Step 3: Determine the slope of the incoming ray
We can rewrite the line equation in slope-intercept form (y = mx + b):
\[ \sqrt{3}y = -x + \sqrt{3} \implies y = -\frac{1}{\sqrt{3}}x + \frac{1}{\sqrt{3}} \]
From this, we see that the slope \( m \) of the incoming ray is:
\[ m = -\frac{1}{\sqrt{3}} \]

### Step 4: Calculate the angle of incidence
The angle of incidence can be found using the slope. The angle \( \theta \) corresponding to the slope \( m \) is given by:
\[ \tan(\theta) = -\frac{1}{\sqrt{3}} \implies \theta = 150^\circ \text{ (since it is in the second quadrant)} \]

### Step 5: Determine the angle of reflection
According to the law of reflection, the angle of reflection is equal to the angle of incidence. Thus, the angle of reflection will also be \( 150^\circ \).

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

### Step 7: Write the equation of the reflected ray
Using the point-slope form of the line equation, where the point is \( A(\sqrt{3}, 0) \) and the slope is \( \frac{1}{\sqrt{3}} \):
\[ y - 0 = \frac{1}{\sqrt{3}}(x - \sqrt{3}) \]
This simplifies to:
\[ y = \frac{1}{\sqrt{3}}x - 1 \]

### Step 8: Rearranging to standard form
To rearrange the equation into standard form:
\[ \sqrt{3}y = x - 1 \implies x - \sqrt{3}y - 1 = 0 \]

Thus, the equation of the reflected ray is:
\[ x - \sqrt{3}y - 1 = 0 \]

### Final Answer
The equation of the reflected ray is:
\[ x - \sqrt{3}y - 1 = 0 \]

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