Let z1 and z2 be two distinct complex numbers and let z=(1−t)z1+tz2 for some real number t with 0<t<1. If arg(ω) denotes the principal argument of a non-zero complex number ω, then
A
|z−z1|+|z+z2|=|z1−z2|
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B
arg(z−z1) = arg(z−z2)
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C
∣∣∣z−z1¯¯¯z−¯¯¯¯¯z1z2−z1¯¯¯¯¯z2−¯¯¯¯¯z1∣∣∣=0
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D
arg(z−z1) = arg(z2−z1)
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Solution
The correct options are A|z−z1|+|z+z2|=|z1−z2| C∣∣∣z−z1¯¯¯z−¯¯¯¯¯z1z2−z1¯¯¯¯¯z2−¯¯¯¯¯z1∣∣∣=0 D arg(z−z1) = arg(z2−z1) Given, z=(1−t)z1+tz2(1−t)t
Clearly, z divides z1 and z2 in the ratio of t : (1-t), 0<t<1 ⇒AP+BP=ABi.e.|z−z1|+|z−z2|=|z1−z2|⇒ Option (a) is true. And arg(z−z1) =arg(z2−z) =arg(z2−z1) ⇒ Option (b) is false and option (d) is true. Also, arg(z−z1)=arg(z2−z1) ⇒arg(z−z1z2−x1)=0 ∴z−z1z2−z1 is purely real. ∴z−z1z2−z1=¯z−¯¯¯¯z1¯¯¯¯z2−¯¯¯¯z1 or ∣∣∣z−z1¯¯¯z−¯¯¯¯¯z1z2−z1¯¯¯¯¯z2−¯¯¯¯¯z1∣∣∣=0 ∴Option (c ) is correct. Hence, ( a, c, d ) is the correct option.