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

Answer 1

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.