SATHEE: Complex Numbers 2 Question 3

Complex Numbers 2 Question 3

3. Let $z_{1}$ and $z_{2}$ be two complex numbers satisfying $| z_{1} | = 9$ and $| z_{2} - 3 - 4 i | = 4$ . Then, the minimum value of $| z_{1} - z_{2} |$ is

(2019 Main, 12 Jan II)

(a) 1

(b) 2

(d) 0

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Answer:

Correct Answer: 3. (d)

Solution:

Clearly $| z_{1} | = 9$ , represents a circle having centre $C_{1} (0, 0)$ and radius $r_{1} = 9$ .

and $| z_{2} - 3 - 4 i | = 4$ represents a circle having centre $C_{2} (3, 4)$ and radius $r_{2} = 4$ .

The minimum value of $| z_{1} - z_{2} |$ is equals to minimum distance between circles $| z_{1} | = 9$ and $| z_{2} - 3 - 4 i | = 4$ .

$∵ C_{1} C_{2} = \sqrt{(3 - 0)^{2} + (4 - 0)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$

and $| r_{1} - r_{2} | = 9 - 4 | = 5 \Rightarrow C_{1} C_{2} \neq r_{1} - r_{2} |$

$∴$ Circles touches each other internally.

Hence, ${| z_{1} - z_{2} |}_{min} = 0$

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Complex Numbers 2 Question 3

3. Let z1 and z2 be two complex numbers satisfying |z1|=9 and |z2−3−4i|=4. Then, the minimum value of |z1−z2| is

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3. Let $z_{1}$ and $z_{2}$ be two complex numbers satisfying $| z_{1} | = 9$ and $| z_{2} - 3 - 4 i | = 4$ . Then, the minimum value of $| z_{1} - z_{2} |$ is