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If z(1),z(2),z(3) are the vertices of an...

If `z_(1),z_(2),z_(3)` are the vertices of an isoscles triangle right angled at `z_(2)`, then

A

`z_(1)^(2)+2z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

B

`z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

C

`z_(1)^(2)+z_(2)^(2)+2z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

D

`2z_(1)^(2)+z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`AB=BC` and `angleABC=pi/2`
`therefore z_(3)-z_(2)=(z_(1)-z_(2))e^(+-ipi//2)`
`rArr (z_(3)-z_(2))=+-i(z_(1)-z_(2))`
`rArr (z_(3)-z_(2))^(2)=-(z_(1)-z_(2))^(2)`
`rArr z_(1)^(2)+2z_(2)^(2)+z_(3)^(2)=2z_(2)(z_(1)+z_(3))`
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