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Let us consider a sector of XOY of a circle of radius r which has been formed by an arc of length S. This arc subtends angle theta radian at the center O of the circle.
Now S/r=theta
=>S=rtheta....[1]
Let the perimeter of the sector P=S+2r......[2]
Combining [1] and [2] we get
P=rtheta +2
=>r=P/(theta+2).....[3]
Now area of the sector A=(pir^2)/(2pi)xxtheta
=>A=r^2/2xxtheta.....[4]
Combining [3] and [4] we get
A=1/2xxP^2/(theta+2)^2xxtheta
Now
A=P^2/2xxtheta/(theta^2+4theta+4)
=>A=P^2/2xx1/(theta+4+4/theta)
=>A=P^2/2xx1/((sqrttheta-2/sqrttheta)^2+8).....[5]
By the given condition the perimeter P of the sector is fixed or constant . So A will be maximum when denominator of the RHS of [5] is minimum and this is possible only when
(sqrttheta-2/sqrttheta)=0
=>theta =2 radian
Hence option (D) is correct
