Let `Z_1 = (8 + i)sin theta + (7 + 4i)cos theta and Z_2 = (1 + 8i)sin theta + (4 + 7i)cos theta` are two complex numbers. If `Z_1* Z_2 = a + ib` where `a, b in R` then the largest value of `(a + b) AA theta in R`, is

Answer

125

Explanation

`Z_(1) = (8sin theta+7 cos theta) + i(sin theta + 4 cos theta) `
` Z_(2) = (sin theta + 4cos )+i(8sin theta + 4cos theta)`
`Z_(2) = (sin theta + 4cos theta) +i(8 sin +4 cos theta)`
Hence, `Z_(1) = x + iy and Z_(2) = y + ix`
where ` x=(8 sin theta + 7 cos theta) and y =(sin theta + 4 cos theta)`
`Z_(1).Z_(2) = (xy -xy) + i(x^(2) +y^(2)) = i(x^(2) + y^(2)) =a+ib`
`rArr a=0, b = x^(2) +y^(2)`
Now, `x^(2) + y^(2) = (8 sin theta + 7 cos theta)^(2) +(sin theta + 4 cos theta)^(2)`
`= 65 sin^(2) theta + 65 cos^(2) theta + 120 sin theta .cos theta`
` = 65 + 60 sin 2 theta`
`rArrZ_(1).Z_(2)|_("max") = 125`