What are the inclination to the X - axis and intercept on Y - axis of the line
`3y =sqrt(3)x+6 `?

Answer

To find the inclination to the X-axis and the intercept on the Y-axis of the line given by the equation \(3y = \sqrt{3}x + 6\), we can follow these steps:

### Step 1: Rewrite the equation in slope-intercept form
We start with the given equation:
\[ 3y = \sqrt{3}x + 6 \]
To convert this into the slope-intercept form \(y = mx + c\), we divide every term by 3:
\[ y = \frac{\sqrt{3}}{3}x + 2 \]

### Step 2: Identify the slope (m) and y-intercept (c)
From the equation \(y = \frac{\sqrt{3}}{3}x + 2\), we can identify:
- The slope \(m = \frac{\sqrt{3}}{3}\)
- The y-intercept \(c = 2\)

### Step 3: Calculate the inclination (θ) to the X-axis
The slope \(m\) is related to the angle of inclination \(\theta\) with the formula:
\[ \tan(\theta) = m \]
Substituting the value of \(m\):
\[ \tan(\theta) = \frac{\sqrt{3}}{3} \]
To find \(\theta\), we take the arctangent:
\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) \]
We know that:
\[ \tan(30^\circ) = \frac{\sqrt{3}}{3} \]
Thus:
\[ \theta = 30^\circ \]

### Conclusion
The inclination of the line to the X-axis is \(30^\circ\) and the intercept on the Y-axis is \(2\) units.