The complex numbers Z - z - z - z form aA- SquareB- RectangleC- RhombusD- None of these

Let z=x+iy⇒z = x iy, z= x iy⇒ z= x+iy AB = |z z| = |2iy| = 2|y| nbsp; BC=|z+z|=2|x| CD = | z+z|=2|y| nbsp; DA = |z+z|=2|x| AC = |z+z|=2|z| BD = |z + z| ...

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Standard XII

Mathematics

Geometrical Representation of a Complex Number

The complex n...

Question

The complex numbers $z, ¯ ¯ ¯ z, - z, - ¯ ¯ ¯ z$ form a

Square

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Rectangle

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Rhombus

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None of these

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Solution

The correct option is B Rectangle

Let $z = x + i y \Rightarrow ¯ ¯ ¯ z = x - i y,$
$- z = - x - i y \Rightarrow - ¯ ¯ ¯ z = - x + i y$

$A B = | z - ¯ ¯ ¯ z | = | 2 i y | = 2 | y |$
$B C = | ¯ ¯ ¯ z + z | = 2 | x |$
$C D = | - z + ¯ ¯ ¯ z | = 2 | y |$
$D A = | z + ¯ ¯ ¯ z | = 2 | x |$
$A C = | z + z | = 2 | z |$
$B D = | ¯ ¯ ¯ z + ¯ ¯ ¯ z | = 2 | ¯ ¯ ¯ z | = 2 | z |$

Here, $A B = C D$ and $B C = D A$
Also, $A C = B D$
Hence, $A, B, C, D$ form a rectangle.

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