about to only mathematics

Answer

2

Explanation

`x^(2)+y^(2)le6 and 2x-3y=1` is shown as

For the point to lie in the shade part, origin and the point lie on opposite side of straight line L.
`therefore` For any point in shaded part `Llt0` and for any point inside the circle `Slt 0`.
Now, for `(2,(3)/(4)) L :2x-3y-1`
`L:4-(9)/(4)-1=(3)/(4)gt0`
`and S:x^(2)+y^(2)-6,S:4+(9)/(16)-6lt0`
`rArr(2,(3)/(4))` lies in shaded part.
For `((5)/(2),(3)/(4)), L:5-9-1lt0" " ("neglect")`
For `((1)/(4),(1)/(4)), L:(1)/(2)+(3)/(4)-1lt0}`
`therefore ((1)/(4),-(1)/(4))` lies in the shaded part.
For `((1)/(8),(1)/(4)), L: (1)/(4),(3)/(4)-1lt0" "["neglect"]`
`rArr` Only 2 points lie in the shaded part.