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

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

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

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

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

Text Solution

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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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Let z_1 and z_2 be two distinct complex numbers and let z= (1-t)z_1+tz_2 for some real number t with 0lttlt1 . If Arg(w) denotes the principal argument of a non-zero complex number w, then

`abs(z-z_1)+abs(z-z_2)=abs(z_1-z_2)`

`Arg(z-z_1)=arg(z-z_2)`

`[(z-z_1),(z_2-z_1)][(barz-barz_1),(barz_2-z_1)]`=0

`Arg(z-z_1)=arg(z_2-z_1)`

If z_(1)and z_(2) are conjugate complex number, then z_(1)+z_(2) will be

real

imaginary

positive integer

nagative integer.

If z_(1) and z_(2) are two complex numbers, then which of the following is true ?

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

`|Z_(1)-Z_(2)| =|Z_(1)| -|Z_(2)|`

`|z_(1)-z_(2)|le|z_(1)|- |z_(2)|`

`|z_(1)+z_(2)|le|z_(1)|+|z_(2)|`