Question

The maximum distance from the origin of coordinates to the point satisfying the equation $|z+\frac{1}{z}|=a$ is

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is C

$letz=r(cos\theta +isin\theta )Then|z+\frac{1}{z}|=a\Rightarrow {|z+\frac{1}{z}|}^2=a^2\Rightarrow r^2+\frac{1}{r^2}+2cos2\theta =a^2.........\left(i\right)$
$Differntiatingw.r.t.\theta weget2r\frac{dr}{d\theta }-\frac{2}{r^3}\frac{dr}{d\theta }-4sin2\theta =0Putting\frac{dr}{d\theta }=0,weget\theta =0,\frac{\pi }{2}rismaximumfor\theta =\frac{\pi }{2},thereforefrom\left(i\right)$
$r^2+\frac{1}{r^2}-2=a^2\Rightarrow r-\frac{1}{r}=a\Rightarrow r=\frac{a+\sqrt{a^2+4}}{2}$