The complex number z Re z -2- satisfies the equation z 2-4 z - z -2-16- z -3- then the value of - z -4 isA- 1B- 4C- 256D- 16

Let z=x+iy x^2 y^2+2ixy=4x+4iy+a, where a is a real number ⇒ 2xy=4y (equating the imaginary part) ⇒ y=0 (∵ x≠2) ⇒ z=x x^2=4x+x^2 16x^3 ⇒ 4x 16x^3=0 ⇒ x^4=4 ⇒ | ...

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Standard XII

Mathematics

Sum of Infinite Terms of a GP

The complex n...

Question

The complex number $z (R e (z) \neq 2)$ satisfies the equation $z^{2} = 4 z + | z |^{2} - \frac{16}{| z |^{3}},$ then the value of $| z |^{4}$ is

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Solution

The correct option is B $4$
Let $z = x + i y$
$x^{2} - y^{2} + 2 i x y = 4 x + 4 i y + a, where a is a real number$
$\Rightarrow 2 x y = 4 y (equating the imaginary part) \Rightarrow y = 0 (∵ x \neq 2) \Rightarrow z = x$

$x^{2} = 4 x + x^{2} - \frac{16}{x^{3}} \Rightarrow 4 x - \frac{16}{x^{3}} = 0 \Rightarrow x^{4} = 4 \Rightarrow | z |^{4} = 4$

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The complex number z (Re(z)≠2) satisfies the equation z2=4z+|z|2−16|z|3, then the value of |z|4 is

The correct option is B 4Let z=x+iy x2−y2+2ixy=4x+4iy+a, where a is a real number ⇒2xy=4y (equating the imaginary part)⇒y=0 (∵x≠2)⇒z=x x2=4x+x2−16x3⇒4x−16x3=0⇒x4=4⇒|z|4=4

The complex number $z (R e (z) \neq 2)$ satisfies the equation $z^{2} = 4 z + | z |^{2} - \frac{16}{| z |^{3}},$ then the value of $| z |^{4}$ is