JEE Mains Maths · Complex Numbers

Regions and Extreme Distances

Sets of complex numbers given by inequalities — disks, half-planes, annuli — and the greatest or least distance from a point to such a set.

All Complex Numbers notes JEE Mains Maths strategy

Why this matters

Twenty-six PYQs, nineteen of them multiple choice, and four from 2026. Seventeen ask for the greatest or least value of a distance such as |z − c| over a circle, a disk or a disk cut by a line; nine describe a region itself — its area, its corners or the lattice points in it. Two ideas cover the page.

Concept 1 of 2: Greatest and least distances

|z-c|

is the distance from

z

to the point

c

. Over a circle with centre

a

and radius

r

it ranges from

\big||c-a|-r\big|

|c-a|+r

, both reached on the line through

c

and

a

. Between two circles whose centres are

d

apart, the greatest distance is

d+r_1+r_2

and, if they are apart, the least is

d-r_1-r_2

. When a line cuts the disk, check whether the extreme point of the whole disk survives the cut; if not, the extreme is at an end of the chord or on the line.

Definition

On $|z-a|=r$ : greatest $|z-c|$ is $|c-a|+r$ , least is $\big||c-a|-r\big|$ .
Over the disk $|z-a|\le r$ : the least is 0 if $c$ is inside.
Two disks with centres $d$ apart: least $\max(0,d-r_1-r_2)$ , greatest $d+r_1+r_2$ .
$|pz+q|=|p|\left|z+\frac qp\right|$ .
Greatest argument on a disk away from 0: the point where a tangent from 0 touches it.

Distance from a point to a circle

\big||c-a|-r\big|\le|z-c|\le|c-a|+r\quad(|z-a|=r)

Worked example

Find the least and greatest

|z-5-12i|

for

|z|=2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2025 · 29 Jan 2025 · Q69Moderate

Example 1 · Complex Numbers · Regions and Extreme Distances

Let

\left| z_{1}- 8 - 2i \right|\leq 1

and

\left| z_{2}- 2 + 6i \right|\leq 2

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

. Then the minimum value of

\left| z_{1}-z_{2} \right|

is :

Is the extreme point still in the region?

For a disk cut by a line, the farthest or nearest point of the whole disk may lie on the removed side. Then the answer sits at an end of the chord or at the foot of a perpendicular on the line.

Drill 16 more on greatest and least distances

Concept 2 of 2: Regions, areas and lattice points

Translate each condition and draw them together.

|z-a|\le r

is a disk and

r_1\le|z-a|\le r_2

an annulus;

|z-a|\le|z-b|

is the half-plane on

a

's side of the bisector; conditions on

\mathrm{Re}

\mathrm{Im}

or a linear expression are half-planes. When the boundary lines pass through the centre of a disk, the region is a sector; a line at distance

p

from the centre cuts off a segment. Counting points with integer coordinates is counting pairs

(u,v)

with

u^2+v^2

in a range.

Definition

$|z-a|\le r$ : disk; $r_1\le|z-a|\le r_2$ : annulus.
$|z-a|\le|z-b|$ : the half-plane containing $a$ .
Sector of angle $\varphi$ : area $\frac12r^2\varphi$ .
Segment cut by a chord subtending $\varphi$ : area $\frac12r^2(\varphi-\sin\varphi)$ .

Sector and segment

\text{sector}=\tfrac12r^2\varphi,\qquad\text{segment}=\tfrac12r^2(\varphi-\sin\varphi)

Worked example

Find the area of

\{z:|z|\le4,\ \mathrm{Re}(z)\ge0,\ \mathrm{Im}(z)\ge0\}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q160Moderate

Example 2 · Complex Numbers · Regions and Extreme Distances

The area (in sq. units) of the region

\mathbf{S = \{ z \in C;|z - 1| \leq 2;(z +}\overline{z}) + i(z -\overline{z}) \leq 2,lm(z) \geq 0\}

Which side of the bisector

|z-a|<|z-b|

is the side that contains

a

. Test one point, such as

a

itself, rather than guessing from the sign of an expanded inequality.

Drill 8 more on regions, areas and lattice points

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Greatest and least distances
Distance from a point to a circle
$\big||c-a|-r\big|\le|z-c|\le|c-a|+r\quad(|z-a|=r)$
Regions, areas and lattice points
Sector and segment
$\text{sector}=\tfrac12r^2\varphi,\qquad\text{segment}=\tfrac12r^2(\varphi-\sin\varphi)$

Watch out for (2)

Is the extreme point still in the region?
→ Greatest and least distances
Which side of the bisector
→ Regions, areas and lattice points

Test yourself on Complex Numbers

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Complex Numbers chapter test Drill all 26 questions