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Let O(0,0),A(2,0),a n dB(1, 1/(sqrt(3)))...

Let `O(0,0),A(2,0),a n dB(1, 1/(sqrt(3)))` be the vertices of a triangle. Let `R` be the region consisting of all those points `P` inside ` O A B` which satisfy `d(P , O A)lt=min[d(P, O B),d(P,A B)]` , where `d` denotes the distance from the point to the corresponding line. Sketch the region `R` and find its area.

A

`sqrt(3)`sq,units

B

`1//sqrt(3)` sq.units

C

`sqrt(3)//2`sq,units

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B

`OP le "min"[BP,AP]`
`OP le AP(whenAPlt BP)`
Let `OP=BP`. The P lies on the perpendicular bisector of OB, For `OP=AP,P` lies on the perpendiuclar bisector of OA. Then, for the required condition, P lies in the region as shown in the diagram. the area of region `OMPN` is
`(1)/(2)xx|{:(0,,0,,),(1,,0,,),(1,,1//sqrt3,,),(1/2,,sqrt(3)//2,,),(0,,0,,):}|=(1)/(2)[(1)/(sqrt3)+(sqrt3)/(2)-(1)/(2sqrt3)]`
`=(1)/(2)[sqrt(3)/(2)+(1)/(sqrt3)]=(1)/(sqrt3)`
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