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P(z(1)),Q(z(2)),R(z(3)) and S(z(4)) are ...

`P(z_(1)),Q(z_(2)),R(z_(3)) and S(z_(4))` are four complex numbers representing the vertices of a rhombus taken in order on the comple plane, then which one of the following is/are correct?

A

`(z_(1)-z_(4))/(z_(2)-z_(3))` is purely real

B

`amp (z_(1) -z_(4))/(z_(2) - z_(4)) = amp (z_(2)-z_(4))/(z_(3) - z_(4))`

C

`(z_(1) - z_(3))/(z_(2)-z_(4))` is pureluy imaginary

D

is not necessary that `|z_(1)-z_(3)|ne |z_(2) - z_(4)|`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D

(1) `PS||QR rArr arg((z_(1)-z_(4))/(z_(2) - z_(3))) = 0 rArr (z_(1) - z_(4))/(z_(2) -z_(4))` is purely real
(2) since diaginal bisect the angle
`rArr amp ((z_(1)-z_(4))/(z_(2)-z_(4)))= amp ((z_(2)-z_(4))/(z_(3) -z_(4)))`
(3) Digonals of rohombus are prependicular. Henc,`(z_(1)-z_(3))//(z_(2)-z_(4))` is purley imagainary .
(4) Digonal of rhombus are not necessarily equal.
Hence, `|z_(1) -z_(3)| ne |z_(2) -z_(4)|` not necessarily .
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