Question

The complex number z satisfies the equation z + |z| = 2 + 8i. Then the value of |z| is

• A. 15
• B. 16
• C. 17
• D. 18

Answer 1

Answer: (C)

17

$z+\left|z\right|=2+8i⟶\left(1\right)$

let $z=x+iy⟶\left(2\right)$

then, $\left|z\right|=\sqrt{x^2+y^2}⟶\left(3\right)$

$\therefore$ Substituting $\left(2\right)$ & $\left(3\right)$ in $\left(1\right)$ we get

$x+iy+\sqrt{x^2+y^2}=2+8i$

$\Rightarrow x+\sqrt{x^2+y^2}+iy=2+8i$

Comparing $LHS$ & $RHS$ we get

$y=8⟶\left(4\right)$ and $x+\sqrt{x^2+y^2}=2⟶\left(5\right)$

Substituting the value of $y$ in the equation $\left(5\right)$

We get

$x+\sqrt{x^2+64}=2$

$\sqrt{x^2+64}=2-x$

Squaring both sides,

$x^2+64=4+x^2-4x$

$\Rightarrow 4x=-60$

$\Rightarrow x=-15$

$\therefore$ $z=-15+8i$

$\left|z\right|=\sqrt{225+164}=\sqrt{289}=17$ units.