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If z = x + iy lies in III quadrant, then...

If z = x + iy lies in III quadrant, then `bar z / z` also lies in III quadrant If:

A

` x gt y gt 0 `

B

`x lt y lt 0`

C

`y lt x lt 0`

D

`y gt x gt 0`

Text Solution

Verified by Experts

The correct Answer is:
B
Given that, z = x + iy lies in third quadrant.
`x lt 0 "and" y lt 0`.
Now. ` (barz)/(z) = (x - iy)/(x + iy)= ((x - iy)( x - iy))/((x + iy)( x + iy))= (x^(2) - y^(2) - 2ixy)/(x^(2) + Y^(2))`
`(barz)/(z) = (x^(2) - Y^(2))/(x^(2) + y^(2)) - (2ixy)/(x^(2)+y^(2))`
Since, `(barz)/(z)` also lies in third quadrant.
` :. = (x^(2) - Y^(2))/(x^(2) + y^(2)) lt 0 "and" (2ixy)/(x^(2)+y^(2))lt 0 `
` x^(2) - y^(2) lt0 "and" -2xy lt0`
`rArr x^(2) lt y^(2) "and" xy gt 0`
So, `x lt y gt 0`
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