If ` A(3/sqrt(2), sqrt(2))`, `B(-3/sqrt(2), sqrt(2)), C(-3/sqrt(2), -sqrt(2))` and `D(3 cos theta , 2 sin theta)` are four points . If the area of the quadrilateral ABCD is maximum where `theta in (3 pi/2, 2 pi)` then (a) maximum area is 10 sq units (b) `theta = 7 pi/4` (c) `theta = 2 pi- sin^(-1) 3/ sqrt(85)` (d) maximum area is 12 sq units

Explanation

Area of qudrilateral AbCD is maximum when area of ACD is maximum. Area of triangle ACD.
`Delta_1=(1)/(2)||{:(3/sqrt2,,sqrt2,,),(3/sqrt2,,-sqrt2,,),(3sintheta,,2costheta,,),(3/sqrt2,,sqrt2,,):}||`
`=|-3sqrt(2)costheta+3sqrt(2)sintheta|`
`therefore Delta_1("max")=6`, when`theta=(7pi)/(4)` (as `theta epsilon(3pi//2,2pi`))
Maximum area is 12 sq.units (as ABCD is a rectangle).