# Locus / Regions in Argand Plane

Since any complex number z = x +iy corresponds to point (x,y) in complex plane (also called**Argand plane**), so many kinds of regions and geometric figures in this plane can be represented by complex equations or inequations.

The equation of circle of radius r and center at origin is |z| = r.

Note that for the circle |z -z_{0}| = r,

all points on the circumference are given
by |z -z_{0}| = r,

all points within the circle are given by |z -z_{0}| < r,

while all points outside the circle are given by |z -z_{0}| > r.

## Illustrative Examples

### Example

What is the equation of the circle in complex plane with radius 2 and center
at 1 +i? Does the origin lie within this circle, on the circle or outside it?
Does any real number lie on this circle?

### Solution

Here center z_{0} = 1 +i, radius = 2.

So the equation of the circle is |z -z_{0}| = r,

that is, |z -(1+i)| = 2

Now, the distance of origin from center

(0 -1)
² +(0 -1)² = 2 < radius 2.

Thus the origin lies within the circle.

Now let the real number (a,0) lie on the circle.

So (a -1)² +(0 -1)² = 2

=> (a -1)² +1 = 4

=> (a -1)² = 3

=> a -1 = ±3

=> a = 1 ±3

Now you try to find out if any purely imaginary number lies on this circle.

### Example

(i) Interpret the loci arg z = /4 in complex plane.

(ii) Interpret the loci arg (z -1) = /4 in complex plane.

(iii) Represent |z +i| = |z -2| in Argand plane.

(iv) How would you represent the line x -y = 0 in terms of complex number z?

### Solution

(i) arg (z) = /4 represents a half ray as shown in the following diagram.

Note that the point z = 0 is not included as arg z is not defined for z = 0.

(ii) arg (z -1) = /4 represents a half ray as shown in following diagram.

Note that the point z = 1 is not included.

(iii) Let z = x +iy, then

|z +i| = |z -2|

=> |x +iy +i| = |x +iy -2|

=> |x +(y +1)i| = |(x -2) +iy|

=> x² +(y +1)² = (x -2)² + y²

=> 4x +2y = 3, which represents a line in Argand plane

Note that intuitively, |z +i| = |z -2| represents all points equidistant from -i and 2 i.e. it represents
the perpendicular bisector of join of -i and 2.

(iv) The line x -y = 0 is shown in the following diagram.

We note taht the points A(-1,i) and B(1,-i) are such that the line x -y = 0 is perpendicular bisector of AB.

Hence required equation in terms of z is

|z -(-1 +i)| = |z -(1 -i)|

i.e. |z +1 -i| = |z -1 +i |.

## Exercise

- Write the equation or inequalities for the following:

(i) a circle of radius 2 with center at origin.

(ii) all points lying outside the circle of radius 2 and center at -1 -i.

(iii) a circle with center at 1 +i and passing through origin.

(iv) all points lying in first or fourth quadrant.

(v) the y-axis.

(vi) all points lying in first quadrant. - Which regions are given by following? Also indicate on complex plane.

(i) 2 |z| 3

(ii) |z -2 -3i| > 2

(iii) Re(z) > 2

(iv) Re(1/z) < 1/2

(v) arg(z -1 -i) = /4

(vi) |z -1| +|z +1| = 3

(vii) |z -1|² +|z +1|² = 4

(viii) |z +i| |z +2|

(ix) |z -2i| = |z +2i| - Find the locus of a complex number z = x +iy satisfying the relation |z +i| = |z +2|. Illustrate the locus of z in the Argand plane.
- Given z
_{1}= 1 +2i. Determine the region in the complex plane represented by 1 < |z -z_{1}|3. Represent it with the help of an Argand diagram. - Find the locus of a complex number z = x +iy satisfying the relation
arg(z -a) = /4, a
**R.**Illustrate the locus of z in Argand diagram. - Find the locus of a complex number z = x +iy satisfying the relation . Illustrate the locus of z in the Argand diagram.

## Answers

**1.**(i) |z| = 2 (ii) |z +1 +i|>2

(iii) |z -1 -i| = 2 [

**Hint.**radius = distance between origin and 1 +i.]

(iv) Re(z)> 0 (v) Re(z) = 0

(vi) Re(z)> 0 and Im(z)> 0

**2.**(i) it represents circular annulus lying between concentric circles of radii 2 and 3 centered at origin. All points on circumference of two circles are included. (You draw the diagram!)

(ii) all points lying outside circle of radius 2 centered at 2 +3i.

(iii) all points to the right of line x = 2.

(iv) exterior of a circle of radius 1 with center (1,0).

(v) portion of a line of inclination 45° passing through point 1 +i of the line x-y = 0.

(vi) an ellipse with foci at (1,0) and (-1,0), and major axis of length 3.

(vii) a circle with points (-1,0), (1,0) as end points of a diameter; in other words, a circle of radius 1 centered at origin.

(viii) all points to the "left" of the line 4x-2y+3=0.

(ix) The x-axis.

**3.**The line 4x-2y+3=0, it is the perpendicular bisector of the join of the points -i and -2. (Please draw the diagram!)

**4.**It represents circular annulus lying between concentric circles of radius 1 and 3 centered at (1,2). The region includes all points on the circumference of the outer circle but excludes all point on the circumference of the inner circle. (You draw the diagram!)

**5.**It represents a portion of the line x-y-a=0. (Please draw the diagram!)

**6.**It represents the circle x² +y² +12 y +4 = 0 with center at (0,-6) and radius = 42 units.