Complex Numbers 2 Question 14
14. For all complex numbers $z _1, z _2$ satisfying $\left|z _1\right|=12$ and $\left|z _2-3-4 i\right|=5$, the minimum value of $\left|z _1-z _2\right|$ is
(a) 0
(b) 2
(c) 7
(d) 17
(2002, 1M)
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Answer:
Correct Answer: 14. (a)
Solution:
- We know, $\left|z _1-z _2\right|=\left|z _1-\left(z _2-3-4 i\right)-(3+4 i)\right|$
$$ \begin{aligned} & \geq\left|z _1\right|-\left|z _2-3-4 i\right|-|3+4 i| \\ & \left.\geq 12-5-5 \quad \text { [using }\left|z _1-z _2\right| \geq\left|z _1\right|-\left|z _2\right|\right] \end{aligned} $$
$$ \therefore \quad\left|z _1-z _2\right| \geq 2 $$
Alternate Solution
Clearly from the figure $\left|z _1-z _2\right|$ is minimum when $z _1, z _2$ lie along the diameter.
$$ \therefore \quad\left|z _1-z _2\right| \geq C _2 B-C _2 A \geq 12-10=2 $$
