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If z(1),z(2)andz(3) are the affixes of t...

If `z_(1),z_(2)andz_(3)` are the affixes of the vertices of a triangle having its circumcentre at the
origin. If zis the affix of its orthocentre, prove that
`Z_(1)+Z_(2)+Z_(3)-Z=0.`

A

`z_(1)+z_(2)+z_(3)+z=0`

B

`z_(1)+z_(2)+z_(3)-z=0`

C

`z_(1)-z_(2)+z_(3)+z=0`

D

`z_(1)+z_(2)-z_(3)+z=0`

Text Solution

Verified by Experts

The correct Answer is:
B
We know that the centroid G, circumference O, and orthocenter `O^(')` of a triangle are collinear such that G divides `OO^(')` in the ratio `1:2`. Since, affix of G is `(z_(1)+z_(2)+z_(3))/3` and O is the origin.

`(z_(1)+z_(2)+z_(3))/(3) = (1 xx z + 2 xx 0)/(1+2)`
`rArr z=z_(1) + z_(2)+z_(3) rArr z_(1)+z_(2)+z_(3)-z=0`
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OBJECTIVE RD SHARMA ENGLISH-COMPLEX NUMBERS -Chapter Test
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