Question

p represents the variable complex number z ,find the locus of p , if arg [z-1/z+3] =

Answer 1

Step-by-step explanation:

Given, arg[

z+1

z−1

]=

3

π

Let z=x+iy

⇒arg[

x+iy+1

x+iy−1

]=

3

π

⇒arg[(x−1)+iy]−arg[(x+1)+iy]=

3

π

⇒tan

−1

[

(x−1)

y

]−tan

−1

[

x+1

y

]=

3

π

⇒tan

−1

⎣

⎢

⎢

⎡

1+

x−1

y

×

x+1

y

x−1

y

−

x+1

y

⎦

⎥

⎥

⎤

=

3

π

⇒

(x−1)(x+1)+y

2

y(x+1)−y(x−1)

=tan

3

π

=

3

⇒

x

2

−1+y

2

y[x+1−x+1]

=

3

⇒

x

2

+y

2

−1

2y

=

3

⇒2y=

3

x

2

+

3

y

2

−

3

Therefore, the locus of P is

3

x

2

+

3

y

2

−2y−

3

=0.

This is an equation of a circle.