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Let z(1) and z(2) be two distinct comple...

Let `z_(1)` and `z_(2)` be two distinct complex numbers and `z=(1-t)z_(1)+tz_(2)`, for some real number t with `0 lt t lt 1` and `i=sqrt(-1)`. If arg(w) denotes the principal argument of a non-zero compolex number w, then

A

`|z-z_(1)| + |z-z_(2)| = |z_(1) -z_(2)|`

B

`(z-z_(1))=(z-z_(2))`

C

`|{:(,z-z_(1),barz-barz_(1)),(,z_(2)-z_(1),barz_(2) -barz_(1)):}|=0`

D

`arg(z-z_(1)) = arg(z_(2) -z_(1))`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
Given `z -=(1-t) z_(1) + tz_(2)`
`rArr z = ((1-t)z_(1) +tz_(2))/((1-t)+t)`
`rArr z` divides the line segment joining `z_(1)` and `z_(2)` in ratio (1-t):t internally as `0 lt t lt 1`
`rArr z,z_(1) and z_(2)` are collinear.
`rArr arg(z-z_(1)) = arg(z_(2) -z)`

`rArr |{:(,z-z_(1),barz - barz_(1)),(,z_(2) - z_(1) , barz_(2) -barz_(1)):}|=0`
`AP + PB = AB `
`rArr |z-z_(1)| + |z-z_(2)|=|z_(1)-z_(2)|`
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