Complex Numbers 2 Question 3
3. Let $z _1$ and $z _2$ be two complex numbers satisfying $\left|z _1\right|=9$ and $\left|z _2-3-4 i\right|=4$. Then, the minimum value of $\left|z _1-z _2\right|$ is
(2019 Main, 12 Jan II)
(a) 1
(b) 2
(c) $\sqrt{2}$
(d) 0
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Answer:
Correct Answer: 3. (d)
Solution:
- Clearly $\left|z _1\right|=9$, represents a circle having centre $C _1(0,0)$ and radius $r _1=9$.
and $\left|z _2-3-4 i\right|=4$ represents a circle having centre $C _2(3,4)$ and radius $r _2=4$.
The minimum value of $\left|z _1-z _2\right|$ is equals to minimum distance between circles $\left|z _1\right|=9$ and $\left|z _2-3-4 i\right|=4$.
$\because C _1 C _2=\sqrt{(3-0)^{2}+(4-0)^{2}}=\sqrt{9+16}=\sqrt{25}=5$
and $\left|r _1-r _2\right|=9-4\left|=5 \Rightarrow C _1 C _2 \neq r _1-r _2\right|$
$\therefore$ Circles touches each other internally.
Hence, $\quad\left|z _1-z _2\right| _{\min }=0$
