Question

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

• A. $\frac{1}{2}$($\sqrt[2]{2(a^2+1)}$+a)
• B. $\frac{1}{2}$($\sqrt[2]{2(a^2+2)}$+a)
• C. $\frac{1}{2}$($\sqrt[2]{2(a^2+4)}$+a)
• D. None of these

Answer 1

The correct option is C

$\frac{1}{2}$($\sqrt[2]{2(a^2+4)}$+a)

let z=r (cos$\theta$+isin$\theta$)

Then $|z+\frac{1}{z}|$=a $\Rightarrow$ ${|z+\frac{1}{z}|}^2$=$a^2$

$\Rightarrow$ $r^2$+$\frac{1}{r^2}$+2cos$\theta$ = $a^2$ $\cdots$ $\cdots$(i)

Differentiating w.r.t $\theta$ we get

2r$\frac{dr}{d\theta }$-$\frac{2}{r^3}$$\frac{dr}{d\theta }$-4sin2$\theta$

Putting $\frac{dr}{d\theta }$=0, we get $\theta$=0,$\frac{\pi }{2}$

r is maximum for $\theta$ = 0, $\frac{\pi }{2}$, therefore from (i)

$r^2+\frac{1}{r^2}-2=a^2\Rightarrow r-\frac{1}{r}$=$a\Rightarrow r$=$\frac{a+\sqrt[2]{2a^2+4}}{2}$