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Consider the traingle having vertices O(...

Consider the traingle having vertices `O(0,0),A(2,0)`, and `B(1,sqrt3)`. Also `b le"min" {a_1,a_2,a_3....a_n}` means ` b le a_1` when `a_1` is least, `b le a_2` when `a_2` is least, and so on. Form this, we can say `b le a_1,b le a_2,.....b le a_n`.
Let R be the region consisting of all those points P inside `DeltaOAB` which satisfy `d(P,OA)le "min"[d(P,OB),d(P,AB)]`, where d denotes the distance from the point to the corresponding line. then the area of the region R is

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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