If `((3)/(2)+(isqrt3)/2)^(50)=3^(25)(x-iy),`
where `x,y in R " and "i=sqrt(-1)`, find the ordered pair of (x,y).

Options

A. (0,3)
B. `(1//2,sqrt(3)//2)`
C. `(-3,0)`
D. `(0,-3)`

Answer

B

Explanation

We have,
`(3/2+(isqrt(3))/(2))^(50)=3^(i25)(x+iy)`
`rArr (isqrt(3))^(50)(1/2-(isqrt(3))/(2))^(50)=3^(25)(x+iy)`
`rArr (-3)^(25)(1/2-(isqrt(3))/(2))^(50)=3^(25)(x+iy)`
`=(1/2-(isqrt(3))/(2))^(50)=(x+iy)`
`rArr -omega^(50)=x+iy`
`rArr -omega^(2)=x+iy`
`rArr 1/2+(isqrt(3))/(2)=x+iy` `[therefore omega^(2)=-1/2-(isqrt(3))/(2)]`
`x=1/2, y=sqrt(3)/2 rArr (x,y)=(1/2,sqrt(3)/2)`