Question

For all complex numbers ${{z}_{1}},{{z}_{2}}$ satisfying $\left| {{z}_{1}} \right|=12$ and $\left| {{z}_{2}}-3-4i \right|=5$, the minimum value of $\left| {{z}_{1}}-{{z}_{2}} \right|$ is
A. $0$
B. $2$
C. \[7\]
D. \[17\]

Answer 1

Hint: In this question, we are to find the minimum value of the given complex subtraction. Here since the required value is minimum, the subtraction takes place between their individual mod amplitudes. By this, we can able to find the required value.



Formula Used:The complex number $(x,y)$ is represented by $x+iy$.
If $z=x+iy\in C$, then $x$ is called the real part and $y$ is called the imaginary part of $z$. These are represented by $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ respectively.
$z=x+iy$ be a complex number such that $\left| z \right|=r$ and $\theta $ be the amplitude of $z$. So, $\cos \theta =\dfrac{x}{r},\sin \theta =\dfrac{b}{r}$
And we can write the magnitude as
 $\begin{align}
  & \left| z \right|=\left| x+iy \right| \\
 & \Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
\end{align}$
Thus, we can write
$z=x+iy=r\cos \theta +ir\sin \theta =r(\cos \theta +i\sin \theta )$
This is said to be the mod amplitude form or the polar form of $z$.
Where $\cos \theta +i\sin \theta $ is denoted by $cis\theta $ and the Euler’s formula is $\cos \theta +i\sin \theta ={{e}^{i\theta }}$
If ${{z}_{1}},{{z}_{2}}$ are two complex numbers, then
$\left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|$; for the minimum value
$\left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|+\left| {{z}_{2}} \right|$; for the maximum value



Complete step by step solution:Given that, ${{z}_{1}},{{z}_{2}}$ are two complex numbers. Such that they satisfy $\left| {{z}_{1}} \right|=12$ and $\left| {{z}_{2}}-3-4i \right|=5$
So, from the given,
\[\begin{align}
  & \left| {{z}_{2}}-3-4i \right|=5 \\
 & \Rightarrow \left| {{z}_{2}}-(3+4i) \right|=5 \\
 & \Rightarrow \left| {{z}_{2}} \right|-\left| 3+4i \right|=5 \\
 & \Rightarrow \left| {{z}_{2}} \right|-\sqrt{{{3}^{2}}+{{4}^{2}}}=5 \\
 & \Rightarrow \left| {{z}_{2}} \right|-5=5 \\
 & \therefore \left| {{z}_{2}} \right|=10 \\
\end{align}\]
So, the minimum value of $\left| {{z}_{1}}-{{z}_{2}} \right|$ is
$\begin{align}
  & \left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right| \\
 & \text{ }=12-10 \\
 & \text{ }=2 \\
\end{align}$



Option ‘B’ is correct



Note: Here we need to remember the condition for the minimum value, the subtraction od complex numbers is in between their individual mod amplitudes i.e., $\left| {{z}_{1}}-{{z}_{2}} \right|=\left| {{z}_{1}} \right|-\left| {{z}_{2}} \right|$. By using this formula, we can find the required minimum value.