SATHEE: Complex Numbers 2 Question 14

Complex Numbers 2 Question 14

14. For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} | = 12$ and $| z_{2} - 3 - 4 i | = 5$ , the minimum value of $| z_{1} - z_{2} |$ is

(a) 0

(b) 2

(c) 7

(d) 17

(2002, 1M)

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Answer:

Correct Answer: 14. (a)

Solution:

We know, $| z_{1} - z_{2} | = | z_{1} - (z_{2} - 3 - 4 i) - (3 + 4 i) |$

$\begin{aligned} \geq | z_{1} | - | z_{2} - 3 - 4 i | - | 3 + 4 i | \\ \geq 12 - 5 - 5 [using | z_{1} - z_{2} | \geq | z_{1} | - | z_{2} |] \end{aligned}$

$∴ | z_{1} - z_{2} | \geq 2$

Alternate Solution

Clearly from the figure $| z_{1} - z_{2} |$ is minimum when $z_{1}, z_{2}$ lie along the diameter.

$∴ | z_{1} - z_{2} | \geq C_{2} B - C_{2} A \geq 12 - 10 = 2$

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Complex Numbers 2 Question 14

14. For all complex numbers z1,z2 satisfying |z1|=12 and |z2−3−4i|=5, the minimum value of |z1−z2| is

Answer:

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14. For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} | = 12$ and $| z_{2} - 3 - 4 i | = 5$ , the minimum value of $| z_{1} - z_{2} |$ is