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Let a, b, c be distinct complex numbers ...

Let `a`, `b`, `c` be distinct complex numbers with `|a|=|b|=|c|=1` and `z_(1)`, `z_(2)` be the roots of the equation `az^(2)+bz+c=0` with `|z_(1)|=1`. Let `P` and `Q` represent the complex numbers `z_(1)` and `z_(2)` in the Argand plane with `/_POQ=theta`, `o^(@) lt 180^(@)` (where `O` being the origin).Then

A

`b^(2)=ac`, `theta=(2pi)/(3)`

B

`theta=(2pi)/(3)`,`PQ=sqrt(3)`

C

`PQ=2sqrt(3)`, `b^(2)=ac`

D

`theta=(pi)/(3)`, `b^(2)=ac`

Text Solution

Verified by Experts

The correct Answer is:
A, B
`(a,b)` `|z_(1)+z_(2)|=|-(b)/(a)|=1`
`|z_(1)z_(2)|=|(c )/(a)|=1`
`:.|z_(2)|=1`
`|z_(1)+z_(2)|^(2)=1`
`:.2+z_(1)barz_(2)+z_(2)barz_(1)=1`
Now `z_(2)=z_(1)e^(itheta)`
`:. |z_(1)+z_(1)e^(itheta)|=|z_(1)||1+e^(itheta)|=1`
`:.2cos"(theta)/(2)=1`
`:.theta=(2pi)/(3)`
Now, `((z_(1)+z_(2))^(2))/(z_(1)z_(2))=1`
`implies (b^(2))/(a^(2))=(c )/(a)`
`impliesb^(2)=ac`
`PQ=|z_(1)-z_(2)|=sqrt(3)`
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