In this section let us study the geometrical interpretation of complex number z in complex plane and the locus of z in Cartesian form.

**Geometry and Locus of Complex Numbers**

In this section let us study the geometrical interpretation of
complex number *z *in complex plane and the locus of *z *in Cartesian
form.

**Example 2.18**

Given the complex
number *z *= 3 + 2*i*, represent the complex numbers *z*, *iz*,
and *z *+ *iz *in one Argand diagram. Show that these complex numbers
form the vertices of an isosceles right triangle.

**Solution**

Given that *z *=
3 + 2*i*.

Therefore, *iz *=
*i *(3 + 2*i *) = -2 + 3*i *

*z *+ *iz *= (3 + 2*i
*) + *i *(3 + 2*i *) = 1+ 5*i*

Let *A*, *B*, and *C *be *z*, *z *+ *iz*,
and *iz *respectively.

AB^{2} = | ( z + iz ) - z|^{2} = |-2 + 3*i*|^{2} = 13

BC^{2} = | iz - ( z + iz )|^{2} = |-3 - 2*i*|^{2} = 13

CA^{2} = | z â€“ iz|^{2} = |5 â€“ *i*|^{2} = 26

Since *AB** ^{2}* +

A circle is defined as the locus of a point which moves in a
plane such that its distance from a fixed point in that plane is always a
constant. The fixed point is the centre and the constant distant is the radius
of the circle.

The locus of *z *that satisfies the equation |z â€“ z_{0}| = r where *z*_{0 } is a fixed
complex number and *r *is a fixed positive real number consists of all
points *z *whose distance from *z*_{0} is *r .*

Therefore |*z *âˆ’ *z*_{0}| = *r *is the complex form of the equation of a circle. (see
Fig. 2.23)

(i) |z - z_{0}| < r represents the points interior of the circle.

(ii) |z - z_{0}| > r represents the points exterior of the circle.

|z| = r â‡’ âˆš[*x*^{2} + *y*^{2}] = *r*

â‡’ *x*^{2} + *y*^{2} = *r*^{2} , represents a circle
centre at the origin with radius *r *units.

Show that |3*z *- 5 + *i* | = 4 represents a
circle, and, find its centre and radius.

The given equation |3*z *- 5 + *i* | = 4 can be written as

It is of the form |z - z_{0}| = r and so it represents a circle, whose
centre and radius are ( 5/3 , - 1/3) and 4/3 respectively.

**Example 2.20**

Show that |*z *+ 2 â€“ *i*|* *< 2 represents
interior points of a circle. Find its centre and radius.

**Solution**

Consider the equation | *z *+ 2 âˆ’*i *| = 2.

This can be written as | *z *âˆ’ (âˆ’2 + *i*)| = 2 .

The above equation represents the circle with centre *z*_{0} = -2 + *i *and
radius *r *= 2. Therefore |*z *+ 2 â€“ *i*|* *< 2
represents all points inside the circle with centre at -2 + *i *and radius
2 as shown in figure.

**Example 2.21**

Obtain the Cartesian form of the locus of *z *in each of
the following cases.

(i) |z| = |z â€“ *i*|

(ii) |2z - 3 â€“ *i*| = 3

**Solution**

(i) we have | z | = |z â€“ *i*|

â‡’ *x* + *iy*|
= |*x + iy* â€“ *i*|

â‡’ âˆš[x^{2} + y^{2}] = âˆš
[*x*^{2} + (*y* â€“ 1)^{2}]

â‡’ x^{2} + y^{2} = *x*^{2} + *y*^{2} - 2y +1

â‡’ 2y -1 = 0 .

(ii) we have |2z - 3 â€“ *i*| = 3

|2 ( *x* + *iy* ) - 3 â€“ *i*| = 3

Squaring on both sides, we get

| (2*x* - 3) + (2*y* -1)*i*|^{2} = 9

â‡’ (2*x* - 3)^{2} + (2*y* -1)^{2} = 9

â‡’ 4*x*^{2} + 4*y*^{2} -12*x* - 4*y*
+1 = 0 , the locus of z in Cartesian form

Tags : Definition, Illustration, Formulas, Solved Example Problems , 12th Mathematics : UNIT 2 : Complex Numbers

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12th Mathematics : UNIT 2 : Complex Numbers : Geometry and Locus of Complex Numbers | Definition, Illustration, Formulas, Solved Example Problems

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