1.

Let Z_(1)=x_(1)+iy_(1), Z_(2)=x_(2)+iy_(2) be complex numbers in fourth quadrant of argand plane and |Z_(1)|=|Z_(2)|=1, Ref(Z_(1)Z_(2))=0. The complex numbers Z_(3)=x_(1)+ix_(2), Z_(4)=y_(1)+iy_(2), Z_(5)=x_(1)+iy_(2), Z_(6)=x_(6)+iy, will always satisfy

Answer»

`|Z_(4)|=1`
`arg(Z_(1)Z_(4))=-pi//2`
`(Z_(5))/(cos(argZ_(1)))+(Z_(6))/(sin(argZ_(1)))` is purely REAL
`Z_(5)^(2)+(barZ_(6))^(2)` is purely imaginergy

Solution :`(a,b,c,d)` `Z_(1)=e^(itheta_(1))`, `Z_(2)=e^(itheta_(2))`, `Re(Z_(1)Z_(2))=0impliestheta_(1)+theta_(2)=-pi//2`, (as `z_(1)`, `z_(2)` lie in fourth quadrant)
`Z_(3)=e^(-itheta_(1))`, `Z_(4)=-e^(itheta_(1))`, `Z_(5)=costheta_(1)(1-i)`, `Z_(6)=sintheta_(1)(-1+i)`


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