Let $\left|\mathrm{z}_{1}-8-2 \mathrm{i}\right| \leq 1$ and $\left|\mathrm{z}_{2}-2+6 \mathrm{i}\right| \leq 2$, $\mathrm{z}_{1}, \mathrm{z}_{2} \in \mathrm{C}$. Then the minimum value of $\left|\mathrm{z}_{1}-\mathrm{z}_{2}\right|$ is :

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Complex Numbers

Let

∣ z_{1} - 8 - 2 i ∣ \leq 1

and

∣ z_{2} - 2 + 6 i ∣ \leq 2

z_{1}, z_{2} \in C

. Then the minimum value of

∣ z_{1} - z_{2} ∣

is :

Answer: (B)

∣ z_{1} - 8 - 2 i ∣ \leq 1

describes the disk of radius

1

centred at

(8, 2)

, and

∣ z_{2} - 2 + 6 i ∣ \leq 2

describes the disk of radius

2

centred at

(2, - 6)

.

The distance between the centres is

(8 - 2)^{2} + (2 - (- 6))^{2} = 36 + 64 = 10.

Since

10

exceeds the sum of the two radii, the disks are disjoint, and the minimum distance between a point of one and a point of the other is

10 - 1 - 2 = 7.

JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Single correct option