LOCUS IN THE COMPLEX PLANE

The locus defines the path of a complex number

Recall from the start of the chapter thatthe modulus and the argument defines

the position of a ‘z’

| a + ib|= a2 +b2

Also recall that the equation of a circle with radius ‘r’ and centre (a,b)

x−a( )2 + y−b( )2 =r2

The complex number z moves in the complex plane subject to the condition Find the equation of the locus of z and interpret the locus geometrically.

| z |=3

z=x+ iy∴ |z|= x2 + y2

|z|=3⇒ x2 + y2

⇒ x2 + y2 =9

The locus of the complex number ‘z’ is therefore the points on the circle centre the origin and radius r = 9

The complex number z moves in the complex plane subject to the condition .Find the equation of the locus of z and interpret the locus geometrically.

⇒

The locus of z is the set of points on the circumference of the circle with centre (1, 2)

and radius 4.

The complex number z moves in the complex plane subject to the condition Find the equation of the locus of z and interpret the locus geometrically.

⇒

The locus of z is the set of points which lie inside the circle with centre (0, 1) and radius 2.

[Note that if the condition was the locus of z would be the set of points which lie on or inside the circle with centre O and radius 2.]

The complex number z moves in the complex plane subject to the condition .Find the equation of the locus of z and interpret the locus geometrically.

The locus of z is the straight line.

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The locus of z is part of the straight line y= 3x x> 0note: when x< 0 z does not have an arguement and the locus is therefor not defined for this part of the line

The complex number z moves in the complex plane such that Show that the locus of z is a straight line and find the equation of the locus of z.

⇒ ⇒⇒⇒⇒⇒

The equation of the locus of z is