Question

If $\left| {z - \dfrac{4}{z}} \right| = 2$, then the maximum value of $\left| z \right|$ is equal to
${\text{A}}{\text{. }}\sqrt 3 + 1 \\\ {\text{B}}{\text{. }}\sqrt 5 + 1 \\\ {\text{C}}{\text{. 2}} \\\ {\text{D}}{\text{. 2 + }}\sqrt 2 \\\$

Answer 1

We have given $\left| {z - \dfrac{4}{z}} \right| = 2$ and we have to find maximum value of $\left| z \right|$ so we have to start from given and use the concept of modulus and complex number so that we can make equation in form of $\left| z \right|$ and then we will be able to find maximum value of $\left| z \right|$.

Complete step by step answer:
We have given
$\left| {z - \dfrac{4}{z}} \right| = 2$
And we have to find $\left| z \right|$
Now we can write $\left| z \right|$= $\left| {z - \dfrac{4}{z} + \dfrac{4}{z}} \right|$
And we know the property of complex number
$\left| {a + b} \right| \leqslant \left| a \right| + \left| b \right|$
We will use this property in above equation as $a = z - \dfrac{4}{z},b = \dfrac{4}{z}$
On applying this property we get,
$\left| z \right| \leqslant \left| {z - \dfrac{4}{z}} \right| + \left| {\dfrac{4}{z}} \right|$
As we have given in question $\left| {z - \dfrac{4}{z}} \right| = 2$ putting this value in above equation we get,
$\left| z \right| \leqslant 2 + \dfrac{4}{{\left| z \right|}}$
On simplifying we get,
$\left| z \right| - 2 - \dfrac{4}{{\left| z \right|}} \leqslant 0$
$\dfrac{{{{\left| z \right|}^2} - 2\left| z \right| - 4}}{{\left| z \right|}} \leqslant 0$
We can write it as also
${\left| z \right|^2} - 2\left| z \right| - 4 = 0$
Here this is quadratic equation in $\left| z \right|$
And for the roots of quadratic equation we know the formula
If $a{x^2} + bx + c = 0$ is the form of quadratic equation then its roots are
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here applying same concept we get,
Hence roots of the quadratic equations are
$z = \pm 2 + \dfrac{{ - \sqrt {4 + 16} }}{2} \\\ = 1 \pm \sqrt 5 \\\$
And hence
$0 \leqslant \left| z \right| \leqslant 1 + \sqrt 5$
And so max $\left| z \right| = 1 + \sqrt 5$
Hence option B is the correct option.

Note:
Whenever we get this type of question the key concept of solving is we have to must remember properties of complex number and modulus like $\left| {a + b} \right| \leqslant \left| a \right| + \left| b \right|$. These help in solving questions easily.