1.

If z1 and z2 are two complex number such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that z1 = - bar z2

Answer»

Given as

|z1| = |z2| and arg (z1) + arg (z2) = π

Let us assume that arg (z1) = θ

arg (z2) = π – θ

As we know that in the polar form, z = |z| (cos θ + i sin θ)

z1 = |z1| (cos θ + i sin θ) …………. (i)

z2 = |z2| (cos (π – θ) + i sin (π – θ))

= |z2| (-cos θ + i sin θ)

= – |z2| (cos θ – i sin θ)

Then let us find the conjugate of

bar z2 = – |z2| (cos θ + i sin θ) …… (ii) (since, |bar z2 = |z2|)

Then,

z1/bar z2 = [|z1| (cos θ + i sin θ)]/[-|z2| (cos θ + i sin θ)]

= – |z1|/|z2| [since, |z1| = |z2|]

= -1

Then, when we cross multiply we get,

z1 = – bar z2
Thus proved.


Discussion

No Comment Found

Related InterviewSolutions