Question

what will be the perimeter of the locus represented by arg(z+i/z-i)=Pie/4 where i=(-1)^(1/2)

Answer 1

its a an arc subtending an angle of pie/4 with i and -i as two end points

Answer 2

can you graph it?

Answer 3

we can graph it only on an argand plane the end points will be i and -i and they will be subtending an angle of pie/4

Answer 4

is your equation
\[ arg(\frac{z+i}{z-i})=\frac{\pi}{4} \]?
because solving arg(z + (i/z) - i) is messy.

Answer 5

you can rewrite (z+i)/(z-i) as (z+z*)/(|z|^2 -1)
so
\[ \frac{z+z^*}{|z|^2 - 1} = tan(\frac{\pi}{4}) =1\]
In terms of real x and imaginary y:
\[ 2x= x^2+y^2-1\]
or
\[ (x-1)^2+y^2= 2 \]
This is the equation of a circle centered at (1,0) with radius sqrt(2). Its perimeter is
\[ 2 \pi r = 2\sqrt{2} \pi \]