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

Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

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.

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 + 2i, 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 + 2i.

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

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

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


AB2 = | ( z + iz ) - z|2 = |-2 + 3i|2 = 13

BC2 = | iz - ( z + iz )|2 = |-3 - 2i|2 = 13

CA2 = | z – iz|2 = |5 – i|2 = 26

Since AB2 + BC2 = CA2 and AB = BC , ΔABC is an isosceles right triangle.

 

Definition 2.5 (circle)

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.

 

Equation of Complex Form of a Circle

The locus of z  that satisfies the equation |z – z0| = r where z0  is a fixed complex number and r is a fixed positive real number consists of all points z whose distance from z0 is r .


Therefore |z z0| = r is the complex form of the equation of a circle. (see Fig. 2.23)

(i) |z - z0| < r represents the points interior of the circle.

(ii) |z - z0| > r represents the points exterior of the circle.

 

Illustration 2.3

|z| = r ⇒ √[x2 + y2] = r 


x2 + y2 = r2 , represents a circle centre at the origin with radius r units.

 

Example 2.19

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

Solution

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


It is of the form |z - z0| = 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 – i| < 2 represents interior points of a circle. Find its centre and radius.

Solution

Consider the equation | z + 2 −i | = 2. 

This can be written as | z − (−2 + i)| = 2 .


The above equation represents the circle with centre z0 = -2 + i and radius r = 2. Therefore |z + 2 – 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 – i|

(ii) |2z - 3 – i| = 3

Solution

(i) we have | z | = |z – i|

 x + iy| = |x + iyi|

⇒ √[x2 + y2] = [x2 + (y – 1)2]

x2 + y2 = x2 + y2 - 2y +1

2y -1 = 0 .

(ii) we have |2z - 3 – i| = 3

|2 ( x + iy ) - 3 – i| = 3

Squaring on both sides, we get

| (2x - 3) + (2y -1)i|2 = 9

(2x - 3)2 + (2y -1)2 = 9

4x2 + 4y2 -12x - 4y +1 = 0 , the locus of z in Cartesian form

 

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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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12th Mathematics : UNIT 2 : Complex Numbers


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