Question

For complex numbers $z_1$, $z_2$ if $\left|z_1\right|=12$ and $\left|z_2-3-4i\right|=5$ then minimum value of $\left|z_1-z_2\right|$ is

• A. $0$
• B. $2$
• C. $7$
• D. $17$

Answer 1

The correct option is B

$2$

Step 1. Given Data,

$\left|z_1\right|=12$

$\left|z_2-3-4i\right|=5$

We have to calculate the minimum value of $\left|z_1-z_2\right|$.

Step 2. We have to find $\left|z_1-z_2\right|$,

Let $z_1=r_1\left(\cos \theta +i\sin \theta \right)$and $z_2=r_2\left(\cos \varphi +i\sin \varphi \right)$.

Hence,

$$\begin{gathered} \left|z_1\right|=\sqrt{{\left(r_1\cos \theta \right)}^2+{\left(r_1\sin \theta \right)}^2} \\ \Rightarrow \left|z_1\right|=r_1 \\ \left|z_2\right|=\sqrt{{\left(r_2\cos \varphi \right)}^2+{\left(r_2\sin \varphi \right)}^2} \\ \Rightarrow \left|z_2\right|=r_2 \end{gathered}$$

$$\begin{gathered} z_1-z_2=r_1\left(\cos \theta +i\sin \theta \right)-r_2\left(\cos \varphi +i\sin \varphi \right) \\ \Rightarrow z_1-z_2=\left(r_1\cos \theta -r_2\cos \varphi \right)+i\left(r_1\sin \theta -r_2\sin \varphi \right) \\ \Rightarrow \left|z_1-z_2\right|=\sqrt{{\left(r_1\cos \theta -r_2\cos \varphi \right)}^2+{\left(r_1\sin \theta -r_2\sin \varphi \right)}^2} \\ \Rightarrow \left|z_1-z_2\right|=\sqrt{r_1^2({\cos }^2\theta +{\sin }^2\theta )+r_2^2({\cos }^2\varphi +{\sin }^2\varphi )-2r_1{r}_2\left(\cos \theta \cos \varphi +\sin \theta \sin \varphi \right)} \\ \Rightarrow \left|z_1-z_2\right|=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos (\theta -\varphi )} \end{gathered}$$

As $\cos (\theta -\varphi )=1$if $\theta =\varphi$

But we have, $\cos (\theta -\varphi )\le 1$

$$\begin{gathered} \Rightarrow \left|z_1-z_2\right|\ge \sqrt{r_1^2+r_2^2-2r_1{r}_2} \\ \Rightarrow \left|z_1-z_2\right|\ge \sqrt{{\left(r_1-r_2\right)}^2} \\ \Rightarrow \left|z_1-z_2\right|\ge r_1-r_2 \\ \Rightarrow \left|z_1-z_2\right|\ge \left|z_1\right|-\left|z_2\right| \end{gathered}$$

Step 3. Calculate the minimum value of $\left|z_1-z_2\right|$

We have, $\left|z_1-z_2\right|\ge \left|z_1\right|-\left|z_2\right|$

Now, we can write $\left|z_2-3-4i\right|$as,

$$\begin{gathered} \left|z_2-3-4i\right|\ge \left|z_2\right|-\left|3+4i\right| \\ \Rightarrow \left|z_2-3-4i\right|\ge \left|z_2\right|-\sqrt{\left(3^2+4^2\right)} \end{gathered}$$

Given that, $\left|z_2-3-4i\right|=5$

Therefore,

$$\begin{gathered} \Rightarrow \left|z_2-3-4i\right|\ge \left|z_2\right|-\sqrt{\left(3^2+4^2\right)} \\ \Rightarrow 5\ge \left|z_2\right|-\sqrt{\left(3^2+4^2\right)} \\ \Rightarrow 5\ge \left|z_2\right|-5 \\ \Rightarrow \left|z_2\right|\ge 10 \end{gathered}$$

Minimum value of $\left|z_1-z_2\right|\ge \left|z_1\right|-\left|z_2\right|$

Given that, $\left|z_1\right|=12$

$$\begin{gathered} \left|z_1-z_2\right|\ge 12-10 \\ \Rightarrow \left|z_1-z_2\right|\ge 2 \end{gathered}$$

Hence, the correct option is $\text{'}B\text{'}$.