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Properties of Argument

If z= x + i...

Question

If $z = x + i y$ then $z ¯ z + 2 (z + ¯ z) + c = 0$ represents

A

a circle

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B

a straight line

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C

a parabola

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D

none of these

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Solution

The correct option is B a circle
$z = x + i y$
$z ¯ z + 2 (z + ¯ z) + c = (x + i y) (x - i y) + 2 (x + i y + x - i y) + c = 0$
$= x^{2} + y^{2} + 2 x + c = 0$
given locus represents a circle.
Hence, option A.

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If z is a non-zero complex number, then $∣ ∣ ∣ \frac{| ¯ z |^{2}}{z ¯ z} ∣ ∣ ∣$ is equal to

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, then show that

z ¯ ¯ ¯ z + 2 (z + ¯ ¯ ¯ z) + a = 0

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∣ ∣ ∣ ∣ \frac{| ¯ z |^{2}}{z ¯ z} ∣ ∣ ∣ ∣

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z ¯ ¯ ¯ z + 2 (z + ¯ ¯ ¯ z) + a = 0

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