JEE Mathematics 2025 Question with Solution

Given: $|z_1 - 8 - 2i| \leq 1$ and $|z_2 - 2 + 6i| \leq 2$.

Find: The minimum value of $|z_1 - z_2|$.

These inequalities represent two discs in the complex plane.

For the first disc,

$$|z_1-(8+2i)|\le 1$$

so its center is $(8,2)$ and radius is $1$.

For the second disc, the solution treats

$$|z_2-2+6i|\le 2$$

as a disc with center $(2,6)$ and radius $2$.

Hence the distance between the centers is

$$\sqrt{(8-2)^2+(2-6)^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$$

Using the minimum-distance-between-two-discs idea,

$$\min |z_1-z_2|=2\sqrt{13}-(1+2)=2\sqrt{13}-3$$

This value is approximately $4.21$, which is not present among the given options.

However, the provided the solution explicitly concludes the final answer as $7$ and marks the correct option as D, while the answer key marks option (4) whose value is $3$. There is a clear discrepancy in the source material.

the defensible answer from the extracted solution is $7$, so the correct option in the given option list is B.

Given: The source solution contains inconsistent interpretation of the second complex inequality.

Find: Which answer should be recorded from the provided source.

In one part of the solution, the centers are written as $(8,2)$ and $(2,6)$ and then the distance between centers is incorrectly taken as $10$.

In another part, the second disc is interpreted as having center $(2,-6)$, giving center distance

$$\sqrt{(8-2)^2+(2-(-6))^2}=\sqrt{36+64}=10$$

and hence minimum distance

$$10-(1+2)=7$$

So both extracted approaches on the page conclude $7$, even though they disagree internally about the center and even though the answer key points to option D, whose printed value is $3$.

Therefore, the provided source is inconsistent. Since the instruction prioritizes the solution, the recorded answer is B.