Q98 / 160 · AP EAMCET 2020 (22 Sep Shift 2) · Mathematics

If $\left|\frac{z-25}{z-1}\right|=5$, then $|z|=$

  • Correct answer
    A
    5
  • B
    3
  • C
    4
  • D
    10

Solution

$\left|\frac{z-25}{z-1}\right|=5$ Let $\quad z=x+i y$ $$ \begin{array}{r} \left|\frac{x+i y-25}{x+i y-1}\right|=5 \\ \frac{|(x-25)+i y|}{|(x-1)+i y|}=5 \\ \frac{\sqrt{(x-25)^2+y^2}}{\sqrt{(x-1)^2+y^2}}=5 \end{array} $$ Squaring on both sides, $$ \begin{aligned} & (x-25)^2+y^2=25\left[(x-1)^2+y^2\right] \\ & \Rightarrow x^2+625-50 x+y^2=25\left[x^2+1-2 x+y^2\right] \\ & \Rightarrow x^2+y^2-50 x+625=25 x^2+25 y^2-50 x+25 \\ & \Rightarrow \quad 24 x^2+24 y^2=600 \end{aligned} $$ $$ \begin{aligned} \Rightarrow & & x^2+y^2 & =25 \\ & \therefore & |z| & =\sqrt{x^2+y^2}=\sqrt{25} \\ & & |z| & =5 \end{aligned} $$ Hence, option (1) is correct.

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

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