complex number question (1 Viewer)

poptarts12345 · Nov 7, 2021

If the complex number z satisfies $\left|z\right|-z-4\left(1-2i\right)=0$ , which of the following is $\left|z^2\right|$ ?

(A) 80
(B) 180
(C) 100
(D) 400

Answers C

ExtremelyBoredUser · Nov 7, 2021

poptarts12345 said:
View attachment 33469
Answers C

let z = x+iy
$\sqrt{x^2+y^2} - x - iy - 4 + 8i = 0$
$\sqrt{x^2+y^2} - x - 4 +i(-y+8)= 0$

Equate real and imaginary parts

$-y + 8 = 0$
$y = 8$

$\sqrt{x^2+y^2} - x - 4 = 0$
$\sqrt{x^2+y^2} = x + 4$
$x^2+y^2 = (x+4)^2$
$x^2+y^2 = x^2 + 8x + 16$
$y^2 = 8x+16$ sub y value in
$x = 6$

Now simply just substitute into definition of |z|^2
$|z|^2 = x^2 + y^2$
$x^2 + y^2 = 100$

or you can just use the equation above and sub x = 6.
$x^2+y^2 = (6+4)^2 = 100$

complex number question (1 Viewer)

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