Question

Let ${{z}_{1}}\text{ and }{{z}_{2}}$ be two complex numbers satisfying $\left| {{z}_{1}} \right|=9\text{ and }\left| {{z}_{2}}-3-4i \right|=4$ then, the minimum value of $\left| {{z}_{1}}-{{z}_{2}} \right|$ is
\[\begin{align}
  & \text{A}.\text{ }0 \\
 & \text{B}.\text{ 1} \\
 & \text{C}.\text{ }\sqrt{2} \\
 & \text{D}.\text{ 2} \\
\end{align}\]

Answer 1

Hint: To solve this question we will first of all use a property of complex number stated below:
\[\left| {{z}_{1}}-{{z}_{2}} \right|\ge \left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\]
We will use the value $\left| {{z}_{2}}-3-4i \right|=4$ in above formula to calculate $\left| {{z}_{2}} \right|$ or least or maximum value of $\left| {{z}_{2}} \right|$. In between this we will use that, if $z=a+ib$ then mode of $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$. After calculating maximum value of $\left| {{z}_{2}} \right|$ we will again apply $\left| {{z}_{1}}-{{z}_{2}} \right|\ge \left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|$ to calculate minimum value of $\left| {{z}_{1}}-{{z}_{2}} \right|$

Complete step-by-step answer:
We have a property of complex number stated as below:
If ${{Z}_{1}}\text{ and }{{Z}_{2}}$ are two complex numbers then \[\left| {{Z}_{1}}-{{Z}_{2}} \right|\ge \left| {{Z}_{1}} \right|-\left| {{Z}_{2}} \right|\]

We are given ${{z}_{1}}\text{ and }{{z}_{2}}$ two complex numbers and $\left| {{z}_{1}} \right|=9\text{ and }\left| {{z}_{2}}-3-4i \right|=4$
Consider \[\left| {{z}_{2}}-3-4i \right|\]
Taking minus common, we get:
\[\left| {{z}_{2}}-\left( 3+4i \right) \right|\]
Using formula stated above by using ${{Z}_{2}}=\left( 3+4i \right)$ we get:
\[\left| {{z}_{2}}-\left( 3+4i \right) \right|\ge \left| {{z}_{2}} \right|-\left| 3+4i \right|\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (i)}\]
We have a complex number $z=a+ib$ then the mode of z is $\left| z \right|=\sqrt{{{a}^{2}}+{{b}^{2}}}$
We have $\left| 3+4i \right|$ applying formula stated above to calculate $\left| 3+4i \right|$ we get:
\[\begin{align}
  & \left| 3+4i \right|=\sqrt{{{\left( 3 \right)}^{2}}+{{\left( 4 \right)}^{2}}} \\
 & \left| 3+4i \right|=\sqrt{9+16} \\
 & \left| 3+4i \right|=\sqrt{25} \\
 & \left| 3+4i \right|=\pm 5 \\
\end{align}\]
Now, as the LHS of the above equation has mode = negative value is not possible.
\[\Rightarrow \left| 3+4i \right|=\pm 5\]
Using this in equation (i), we get:
\[\left| {{z}_{2}}-\left( 3+4i \right) \right|\ge \left| {{z}_{2}} \right|-5\]
Now, as \[\begin{align}
  & \left| {{z}_{2}}-\left( 3+4i \right) \right|=4 \\
 & \Rightarrow 4\ge \left| {{z}_{2}} \right|-5 \\
\end{align}\]
Adding 5 both sides, we get:
\[\begin{align}
  & 4+5\ge \left| {{z}_{2}} \right| \\
 & \Rightarrow \left| {{z}_{2}} \right|\le 9\text{ }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. }\text{. (ii)} \\
\end{align}\]
Now, we have for two complex number ${{z}_{1}}\text{ and }{{z}_{2}}$
\[\left| {{z}_{1}}-{{z}_{2}} \right|\ge \left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|\]
Then, minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-{{\left| {{z}_{2}} \right|}_{\max }}\]
Minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-{{\left| {{z}_{2}} \right|}_{\max }}\] from equation (ii) we have ${{\left| {{z}_{2}} \right|}_{\max }}=9$ and given in question is $\left| {{z}_{1}} \right|=9$
Minimum value of \[\Rightarrow \left| {{z}_{1}}-{{z}_{2}} \right|=9-9=0\]
Therefore, the minimum value of \[\left| {{z}_{1}}-{{z}_{2}} \right|=0\] so

So, the correct answer is “Option A”.

Note: If a number is of the form $\left| z \right|\le a$ then minimum value of $\left| z \right|$ is not known but maximum value of $\left| z \right|=a$ this is what we have used for $\max \left| {{z}_{2}} \right|\Rightarrow \left| {{z}_{2}} \right|\le 9$ So, $\max \left( \left| {{z}_{2}} \right| \right)=9$
A key point to note is, at the end of solution where we have used \[\min \left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-{{\left| {{z}_{2}} \right|}_{\max }}\]
This is so as if we have a difference of two numbers x-y=a then as larger the value of y, we get the least value of a because subtracting larger value gives least a.
Therefore, the step \[\min \left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-{{\left| {{z}_{2}} \right|}_{\max }}\] is correct.