The straight line 2x-3y = 1 divides the circular region `x^2+ y^2 le6` into two parts. If S = { `( 2 , 3/4) , (5/2,3/4) , (1/4,-1/4), (1/8,1/4)`}, then the number of point(s) in S lying inside the smaller part is

Explanation

To solve the problem, we need to determine how many points from the set \( S = \{ (2, \frac{3}{4}), (\frac{5}{2}, \frac{3}{4}), (\frac{1}{4}, -\frac{1}{4}), (\frac{1}{8}, \frac{1}{4}) \} \) lie inside the smaller part of the circular region defined by \( x^2 + y^2 \leq 6 \) that is divided by the line \( 2x - 3y = 1 \).

### Step 1: Identify the Circle and Line
The circle is given by the equation \( x^2 + y^2 = 6 \), which has its center at the origin (0, 0) and a radius of \( \sqrt{6} \).

The line is given by the equation \( 2x - 3y = 1 \). We can rearrange this to find the y-intercept:
\[ y = \frac{2}{3}x - \frac{1}{3} \]

### Step 2: Determine the Side of the Line
To find out which side of the line the points lie on, we substitute the points into the line equation \( L = 2x - 3y - 1 \).

### Step 3: Check Each Point
1. **Point \( (2, \frac{3}{4}) \)**:
\[ L = 2(2) - 3\left(\frac{3}{4}\right) - 1 = 4 - \frac{9}{4} - 1 = 4 - 2.25 - 1 = 0.75 > 0 \]
Now check if it lies inside the circle:
\[ S = 2^2 + \left(\frac{3}{4}\right)^2 - 6 = 4 + \frac{9}{16} - 6 = 4 + 0.5625 - 6 = -1.4375 < 0 \]
This point lies in the smaller region.

2. **Point \( (\frac{5}{2}, \frac{3}{4}) \)**:
\[ L = 2\left(\frac{5}{2}\right) - 3\left(\frac{3}{4}\right) - 1 = 5 - \frac{9}{4} - 1 = 5 - 2.25 - 1 = 1.75 > 0 \]
Now check if it lies inside the circle:
\[ S = \left(\frac{5}{2}\right)^2 + \left(\frac{3}{4}\right)^2 - 6 = \frac{25}{4} + \frac{9}{16} - 6 = \frac{25}{4} + 0.5625 - 6 \]
Converting \( 6 \) to quarters:
\[ = \frac{25}{4} + \frac{9}{16} - \frac{24}{4} = \frac{25 - 24}{4} + \frac{9}{16} = \frac{1}{4} + \frac{9}{16} = \frac{4}{16} + \frac{9}{16} = \frac{13}{16} > 0 \]
This point does not lie in the smaller region.

3. **Point \( (\frac{1}{4}, -\frac{1}{4}) \)**:
\[ L = 2\left(\frac{1}{4}\right) - 3\left(-\frac{1}{4}\right) - 1 = \frac{1}{2} + \frac{3}{4} - 1 = \frac{1}{2} + \frac{3}{4} - 1 = \frac{2}{4} + \frac{3}{4} - \frac{4}{4} = \frac{1}{4} > 0 \]
Now check if it lies inside the circle:
\[ S = \left(\frac{1}{4}\right)^2 + \left(-\frac{1}{4}\right)^2 - 6 = \frac{1}{16} + \frac{1}{16} - 6 = \frac{2}{16} - 6 = \frac{1}{8} - 6 < 0 \]
This point lies in the smaller region.

4. **Point \( (\frac{1}{8}, \frac{1}{4}) \)**:
\[ L = 2\left(\frac{1}{8}\right) - 3\left(\frac{1}{4}\right) - 1 = \frac{1}{4} - \frac{3}{4} - 1 = -\frac{2}{4} - 1 = -\frac{1}{2} < 0 \]
Now check if it lies inside the circle:
\[ S = \left(\frac{1}{8}\right)^2 + \left(\frac{1}{4}\right)^2 - 6 = \frac{1}{64} + \frac{1}{16} - 6 = \frac{1}{64} + \frac{4}{64} - 6 = \frac{5}{64} - 6 < 0 \]
This point does not lie in the smaller region.

### Conclusion
The points that lie inside the smaller part of the circle are:
- \( (2, \frac{3}{4}) \)
- \( (\frac{1}{4}, -\frac{1}{4}) \)

Thus, the number of points in \( S \) lying inside the smaller part is **2**.