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If the points A(z),B(-z),C(1-z) are the ...

If the points A(z),B(-z),C(1-z) are the vertices of an equilateral triangle ABC then Re(z) is

A

`tan^(-1)((sqrt(15))/(5))`

B

`tan^(-1)(sqrt(15))`

C

`tan^(-1)((5)/(sqrt(15)))`

D

`(pi)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `M(z')` is mid point of `AB`, so `z'=(z+2)/(2)`

`impliesbarz'=(barz+2)/(2)`
Now `z`, `1`,`(barz)/(2)+1` are collinear
`implies|{:(z,barz,1),((barz)/(2)+1,(z)/(2)+1,1),(1,1,1):}|=0`
`impliesz((z)/(2)+1-1)-barz((barz)/(2)+1-1)+1((barz)/(2)-(z)/(2))=0`
`implies(z^(2))/(2)-(barz)^(2)/(2)+(barz-z)/(2)=0`
`implies(z-barz)(z+barz-1)=0`
`impliesz-barz=0` or `(z+barz-1)=0`
`impliesz+barz=1` or `Re(z)=(1)/(2)`
`|z|=2implies(1)/(4)+(Im(z))^(2)=4`
`impliesIm(z)=(sqrt(15))/(2)`
`impliesz=(1)/(2)+(isqrt(15))/(2)`
`impliesarg(z')=tan^(-1)((sqrt(15))/(5))`
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