Let the circles and , , be such that lies within . If moves on , moves on and , then is equal to: [2026]

Question

Let the circles $C_{1} : | z | = r$ and $C_{2} : | z - 3 - 4 i | = 5$ , $z \in C$ , be such that $C_{2}$ lies within $C_{1}$ . If $z_{1}$ moves on $C_{1}$ , $z_{2}$ moves on $C_{2}$ and $\min | z_{1} - z_{2} | = 2$ , then $\max | z_{1} - z_{2} | = 2$ is equal to: [2026]

Smart Achievers · Accepted Answer

(3)

We have, |z| = r, Centre is (0, 0); radius = r

|z – 3 – 4i| = 5

Centre is (3, 4); radius = 5

Distance between centres $= \sqrt{3^{2} + 4^{2}} = 5$

$C_{2}$ lies inside $C_{1}$ and $| z_{1} - z_{2} |_{\min} = 2$

For internal circle, minimum distance between points on the two circles is r – 5 – 5 = r –10

So, r – 10 = 2 $\Rightarrow$ r = 12

Maximum distance between a point on $C_{1}$ and a point on $C_{2}$ : r + 5 + 5 = 12 + 10 = 22

$∴ | z_{1} - z_{2} |_{\max} = 22$

Solution

Q.
Let the circles $C_{1} : | z | = r$ and $C_{2} : | z - 3 - 4 i | = 5$ , $z \in C$ , be such that $C_{2}$ lies within $C_{1}$ . If $z_{1}$ moves on $C_{1}$ , $z_{2}$ moves on $C_{2}$ and $\min | z_{1} - z_{2} | = 2$ , then $\max | z_{1} - z_{2} | = 2$ is equal to: [2026]

1 12

2 17

3 22

4 24

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