If z and are two non-zero complex numbers such that |z | = 1, and Arg(z) - Maths - Complex Numbers and Quadratic Equations

Question

If z and ω are two non-zero complex numbers such that |z
ω| = 1, and Arg(z) – Arg( ω) = π/2, the z
ω is equal to
(1) 1 (2) –1 (3) i (4) –i

Shruti Tyagi · Accepted Answer

Dear Student,

if z1,z2 and z3 are vertices of an equilateral triangle thenz21+z22+z23= z1×z2 +z2×z3 +z1×z3 let origin,z1 and z2 are vertices of triangle thenz21+z22+02=z1×z2+0×z1+0×z2⇒z21+z22=z1×z2⇒z21+z22+2z1×z2-2z1×z2=z1×z2⇒z21+z22+2z1×z2=3z1×z2⇒z1+z22=3z1×z2---1again z1 and z2  are roots of equation z2+az+b=0Hence z1+z2=-aand z1×z2=bputting values in equation 1⇒-a2=3b⇒a2=3b

Regards,

If z and ω are two non-zero complex numbers such that |z ω| = 1, and Arg(z) – Arg( ω) = π/2, the z ω is equal to (1) 1 (2) –1 (3) i (4) –i

If z and ω are two non-zero complex numbers such that |z ω| = 1, and Arg(z) – Arg( ω) = π/2, the $z$ ω is equal to
(1) 1 (2) –1 (3) i (4) –i