Verified Solution • 2025–26 Session moderate

QJEEM05Ma281a: JEE Main 2005 Mathematics - Class 11 JEEM05Ma281a Solution: If and are two non-zero complex numbers such that , then arg -…

If z1{z_1} and z2{z_2} are two non-zero complex numbers such that  ∣z1+z2∣=∣z1∣+∣z2∣\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right| , then arg z1{z_1} - arg z2{z_2} is equal to :

Multiple Choice Options

  • A π2 {\pi \over 2}\,
  • B −π- \pi
  • C 0
  • D −π2{{ - \pi } \over 2}

Step-by-Step Verified Solution

Given that,  ∣z1+z2∣=∣z1∣+∣z2∣\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|

 ∣z1+z2∣\,\left| {{z_1} + {z_2}} \right| is the vector sum of z1{z_1} and z2{z_2} . So  ∣z1+z2∣\,\left| {{z_1} + {z_2}} \right| should be < ∣z1∣+∣z2∣\left| {{z_1}} \right| + \left| {{z_2}} \right| but here they are equal so z1{z_1} and z2{z_2} are collinear.

S if z1{z_1} makes an angle θ\theta with x axis then z2{z_2} will also make θ\theta angle.

∴\therefore arg z1{z_1} - arg z2{z_2} = θ\theta - θ\theta = 0

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