JEE Mains Maths · Complex Numbers

Lines and Circles in the Complex Plane

Reading an equation in z as a straight line or a circle: equal distances from two points, linear equations in z and its conjugate, and circles given by a centre or by a ratio of distances.

All Complex Numbers notes JEE Mains Maths strategy

Why this matters

Seventeen PYQs, twelve of them multiple choice. Seven are straight lines — perpendicular bisectors and linear equations in z and its conjugate; ten are circles, often a ratio of two distances. Most then intersect the line or circle with another curve. Two ideas cover the page.

Concept 1 of 2: Straight lines

|z-a|=|z-b|

says

z

is equally far from

a

and

b

: the perpendicular bisector of the segment

ab

. A linear equation

\bar az+a\bar z+c=0

with

c

real is a line, because

\bar az+a\bar z=2\,\mathrm{Re}(\bar az)

is linear in

x

and

y

. So is

|z-a|^2-|z-b|^2=k

, since the

x^2+y^2

terms cancel. To meet a circle, substitute the line into it.

Definition

$|z-a|=|z-b|$ : the perpendicular bisector of $a$ and $b$ .
$\bar az+a\bar z+c=0$ , $c$ real: a line.
$z(p+iq)+\bar z(p-iq)=2(px-qy)$ .
$|z-a|^2-|z-b|^2=k$ : a line at right angles to $ab$ .

Line in complex form

\bar az+a\bar z+c=0\quad(c\in\mathbb R)

Worked example

Write

|z-1|=|z+3i|

x

and

y

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 25 Jan 2023 · Q65Moderate

Example 1 · Complex Numbers · Lines and Circles in the Complex Plane

Let

z_{1}= 2 + 3i

and

z_{2}= 3 + 4i

. The set

S =\left\{ z \in C:\left| z -z_{1} \right|^{2}-\left| z -z_{2} \right|^{2}=\left| z_{1}-z_{2} \right|^{2} \right\}

represents a

Expand before trusting a sign

z(1+i)+\bar z(1-i)=2(x-y)

, while

z(1-i)+\bar z(1+i)=2(x+y)

. Expanding once with

z=x+iy

avoids picking the wrong half-plane or line.

Drill 6 more on straight lines

Concept 2 of 2: Circles

|z-a|=r

is the circle with centre

a

and radius

r

. A ratio of distances,

|z-a|=k|z-b|

with

k\neq1

, is also a circle: square, expand, and divide by the coefficient of

x^2+y^2

before reading off the centre. In complex form,

z\bar z+\bar\alpha z+\alpha\bar z+d=0

is the circle with centre

-\alpha

and radius

\sqrt{|\alpha|^2-d}

Definition

$|z-a|=r$ : centre $a$ , radius $r$ .
$|z-a|=k|z-b|$ , $k\neq1$ : a circle (for $k=1$ , a line).
$z\bar z+\bar\alpha z+\alpha\bar z+d=0$ : centre $-\alpha$ , $r^2=|\alpha|^2-d$ .
$|z-\alpha|^2+|z-\beta|^2=2|z-m|^2+\frac12|\alpha-\beta|^2$ , $m=\frac{\alpha+\beta}2$ .

Circle in complex form

z\bar z+\bar\alpha z+\alpha\bar z+d=0:\ \text{centre }-\alpha,\ r=\sqrt{|\alpha|^2-d}

Worked example

Find the centre and radius of

|z+2|=3|z-2|

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 1 February 2023 · Q78Moderate

Example 2 · Complex Numbers · Lines and Circles in the Complex Plane

If the center and radius of the circle

\left| \frac{z - 2}{z - 3} \right|= 2

are respectively

(\alpha,\beta)

and

\gamma

. then

3(\alpha + \beta + \gamma)

is equal to

Divide before reading the centre

After squaring a ratio of distances the

x^2+y^2

term has a coefficient like 3 or 8. Divide through by it first; reading the centre from the undivided equation gives the wrong point.

Drill 9 more on circles

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)

Straight lines
Line in complex form
$\bar az+a\bar z+c=0\quad(c\in\mathbb R)$
Circles
Circle in complex form
$z\bar z+\bar\alpha z+\alpha\bar z+d=0:\ \text{centre }-\alpha,\ r=\sqrt{|\alpha|^2-d}$

Watch out for (2)

Expand before trusting a sign
→ Straight lines
Divide before reading the centre
→ Circles

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 17 questions