Q47 / 160 · TS EAMCET 2020 (14 Sep Shift 2) · Mathematics

Let $z$ be a complex number such that $|z|-z=2+i$, where $i=\sqrt{-1}$. Then, $|z|=$

  • A
    $\frac{5}{2}$
  • B
    $\frac{\sqrt{41}}{4}$
  • C
    $\frac{5}{3}$
  • Correct answer
    D
    $\frac{5}{4}$

Solution

We have, $|z|-z=2+i$ Let $z=x+i y \quad \therefore \sqrt{x^2+y^2}-x-i y=2+i$ Equating real part and imaginary part we get, $\begin{array}{lll} & \sqrt{x^2+y^2}-x=2 \text { and } y=-1 \\ \therefore & \sqrt{x^2+1}=x+2 \Rightarrow x^2+1=x^2+4 x+4 \\ \Rightarrow & x=-\frac{3}{4} \Rightarrow z=-\frac{3}{4}-i \\ \therefore & |z|=\sqrt{\frac{9}{16}+1}=\sqrt{\frac{25}{16}}=\frac{5}{4} \end{array}$

Hence, the correct answer is option: 'D'.

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