Question

If $z\in C$, then $\left|\frac{2z-i}{5z+1}\right|=m,m\in R$ represents a straight line if

• A. $m=2$
• B. $m=5$
• C. $m=2.5$
• D. $m=0.4$

Answer 1

The correct option is D $m=0.4$

Simplifying, we get
$\frac{2}{5}\left|\frac{z-\frac{i}{2}}{z+\frac{1}{5}}\right|=m$
$\left|\frac{z-\frac{i}{2}}{z+\frac{1}{5}}\right|=\frac{5m}{2}$
$\left|\frac{z-\frac{i}{2}}{z+\frac{1}{5}}\right|=2.5\left(m\right)$
For ${}^{\prime }{z}^{\prime }$ to lie on a straight line
$2.5m=1$
$m=\frac{2}{5}$
$m=0.4$
Hence, option 'D' is correct.