Q16 of 66 Page 91

z1 and z2 are two complex numbers such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that

Let z1 = |z1| (cos θ1 + I sin θ1) and z2 = |z2| (cos θ2 + I sin θ2)


Given that |z1| = |z2|


And arg (z1) + arg (z2) = π


⇒ θ1 + θ2 = π


⇒ θ1 = π – θ2


Now, z1 = |z2| (cos (π - θ2) + I sin (π - θ2))


⇒ z1 = |z2| (-cos θ2 + I sin θ2)


⇒ z1 = -|z2| (cos θ2 – I sin θ2)


⇒ z1 = - [|z2| (cos θ2 – I sin θ2)]


∴ z1 = -z̅ 2


Hence proved.


More from this chapter

All 66 →
14

Show that represents a circle. Find its centre and radius.

 15

If is a purely imaginary number (z ≠ –1), then find the value of |z|.

 17

If |z1| = 1 (z1 ≠ –1) and then show that the real part of z2 is zero.

 18

If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find