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

Question

Aicharya · Accepted Answer

$\left|\frac{z-25}{z-1}ight|=5$ Let $\quad z=x+i y$ $ \begin{array}{r} \left|\frac{x+i y-25}{x+i y-1}ight|=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ight] \ & \Rightarrow x^2+625-50 x+y^2=25\left[x^2+1-2 x+y^2ight] \ & \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 \ & 	herefore & |z| & =\sqrt{x^2+y^2}=\sqrt{25} \ & & |z| & =5 \end{aligned} $ Hence, option (1) is correct

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

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