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Question
If z = x + iy lies in the third quadrant, then `barz/z` also lies in the third quadrant if ______.
Options
x > y > 0
x < y < 0
y < x < 0
y > x > 0
Solution
If z = x + iy lies in the third quadrant, then `barz/z` also lies in the third quadrant if x < y < 0.
Explanation:
Given that: z = x + iy
If z lies in third quadrant.
So x < 0 and y < 0.
`barz` = x – iy
`barz/z = (x - iy)/(x + iy)`
= `(x - iy)/(x + iy) xx (x - iy)/(x -iy)`
= `(x^2 + i^2y^2 - 2xyi)/(x^2 - i^2y^2)`
= `(x^2 - y^2 - 2xyi)/(x^2 + y^2)`
= `(x^2 - y^2)/(x^2 + y^2) - (2xy)/(x^2 + y^2) i`
When z lies in third quadrant then `barz/z` will also be lie in third quadrant.
∴ `(x^2 - y^2)/(x^2 + y^2) < 0` and `(-2xy)/(x^2 + y^2) < 0`
⇒ x2 – y2 < 0 and 2xy > 0
⇒ x2 < y2 and xy > 0
So x < y < 0.
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