a ray of light along x+root3 y =root 3  get reflected upon reaching x-axis, the equation of the reflected ray is

To find the equation of the reflected ray when the ray of light described by the equation \( x + \sqrt{3}y = \sqrt{3} \) reflects off the x-axis, follow these steps:

### 1. **Identify the original equation**:
The original equation of the ray is:
\[
x + \sqrt{3}y = \sqrt{3}
\]

### 2. **Rewrite the equation in slope-intercept form**:
To make the reflection process easier, let's first convert the equation into the slope-intercept form \( y = mx + b \).

\[
\sqrt{3}y = -x + \sqrt{3}
\]
\[
y = -\frac{1}{\sqrt{3}}x + 1
\]
\[
y = -\frac{\sqrt{3}}{3}x + 1
\]

So, the slope (\(m\)) of the original ray is \( -\frac{\sqrt{3}}{3} \), and the y-intercept (\(b\)) is 1.

### 3. **Determine the slope of the reflected ray**:
When a ray is reflected off the x-axis, the slope of the reflected ray changes sign (i.e., the slope becomes the negative of the original slope). Therefore, the slope of the reflected ray will be:
\[
m_{\text{reflected}} = \frac{\sqrt{3}}{3}
\]

### 4. **Find the equation of the reflected ray**:
Using the slope of the reflected ray and the point where the ray reflects, which is the same as the x-intercept of the original ray, we can write the equation of the reflected ray.

The reflected ray will pass through the same point on the x-axis as the original ray. To find this point, set \( y = 0 \) in the original equation \( y = -\frac{\sqrt{3}}{3}x + 1 \).

\[
0 = -\frac{\sqrt{3}}{3}x + 1
\]
\[
\frac{\sqrt{3}}{3}x = 1
\]
\[
x = \frac{3}{\sqrt{3}} = \sqrt{3}
\]

So, the point of reflection is \( (\sqrt{3}, 0) \).

Now, using the point-slope form to write the equation of the reflected ray:
\[
y - 0 = \frac{\sqrt{3}}{3}(x - \sqrt{3})
\]
\[
y = \frac{\sqrt{3}}{3}x - 1
\]

### 5. **Final equation**:
The equation of the reflected ray is:
\[
y = \frac{\sqrt{3}}{3}x - 1
\]

This is the equation of the ray after it reflects off the x-axis.

Would you like further explanation or have any questions on this topic?

### Relative Questions:
1. What happens to a ray of light reflecting off the y-axis?
2. How do you find the point of intersection between a line and the x-axis?
3. How would the equation change if the ray reflected off the line \(y = -x\)?
4. What is the general formula for reflection of a point over the x-axis?
5. How can you verify that the reflection is correct geometrically?

### Tip:
When reflecting a line over the x-axis, simply change the sign of the slope, and adjust the intercept to ensure the line passes through the same point on the x-axis.

Explore how to find the equation of a reflected ray of light off the x-axis. Learn step-by-step with an example involving coordinate geometry and the slope-intercept form. Suitable for high school students studying physics or coordinate geometry.

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