1. Circle and dividing line

The circular region is $x^2+y^2\le 6,$ so the circle has center $(0,0)$ and radius $\sqrt{6}$.

The line is $2x-3y=1.$ This line divides the disk into two parts. Since the center $(0,0)$ does not lie on the line, the smaller part is the side of the line away from the center.

Let us check which side contains the center: $2(0)-3(0)-1=-1<0.$ So the center lies in the region $2x-3y-1<0.$ Hence the smaller part is the other side: $2x-3y-1>0.$

Also, since all given points must lie inside the circular region as well, we will check both:

• $x^2+y^2\le 6$
• $2x-3y-1>0$

2. Given set of points

The set is $S=\left\{\left(2,\frac34\right),\left(\frac52,\frac34\right),\left(\frac14,-\frac14\right),\left(\frac18,\frac14\right)\right\}.$

3. Test each point

(i) Point $\left(2,\frac34\right)$

Check circle: $x^2+y^2=2^2+\left(\frac34\right)^2=4+\frac{9}{16}=\frac{73}{16}<6$ since $6=\frac{96}{16}.$ So it lies inside the disk.

Check line side: $2x-3y-1=2(2)-3\left(\frac34\right)-1=4-\frac94-1=3-\frac94=\frac34>0.$ So this point lies in the smaller part.

(ii) Point $\left(\frac52,\frac34\right)$

Check circle: $x^2+y^2=\left(\frac52\right)^2+\left(\frac34\right)^2=\frac{25}{4}+\frac{9}{16}=\frac{100}{16}+\frac{9}{16}=\frac{109}{16}.$ Now, $6=\frac{96}{16},$ so $\frac{109}{16}>6.$ Hence this point is outside the circular region, so it cannot lie inside either part.

(iii) Point $\left(\frac14,-\frac14\right)$

Check circle: $x^2+y^2=\left(\frac14\right)^2+\left(-\frac14\right)^2=\frac1{16}+\frac1{16}=\frac18<6.$ So it lies inside the disk.

Check line side: $2x-3y-1=2\left(\frac14\right)-3\left(-\frac14\right)-1=\frac12+\frac34-1=\frac54-1=\frac14>0.$ So this point also lies in the smaller part.

(iv) Point $\left(\frac18,\frac14\right)$

Check circle: $x^2+y^2=\left(\frac18\right)^2+\left(\frac14\right)^2=\frac1{64}+\frac1{16}=\frac1{64}+\frac4{64}=\frac5{64}<6.$ So it lies inside the disk.

Check line side: $2x-3y-1=2\left(\frac18\right)-3\left(\frac14\right)-1=\frac14-\frac34-1=-\frac12-1=-\frac32<0.$ So this point lies in the larger part, not the smaller part.

4. Count points in the smaller part

Points in the smaller part are: $\left(2,\frac34\right),\quad \left(\frac14,-\frac14\right).$

Thus, the number of points is $\boxed{2}.$

5. Comparison with stored answer

Stored correct answer = $2$.

Our derived answer is also $2$, so they agree.