Complex Numbers Part 1

arg(

z−2

z+2

)=

π

4

Ifz=x+iy
arg[(x−2)+iy]−arg[(x+2)+iy]=

π

4

⇒tan−1(

y

x−2

)−tan−1(

y

x+2

)=

π

4

⇒

y x−2 − y x+2

1+ y x−2 . y x+2

=tan(

π

4

)
⇒

xy+2y−xy+2y

x2+y2−4

=1
⇒4y=x2+y2−4
⇒x2+y2−4y−4=0
z is a circle.
Centre =(0,2), Radius =(2√2)
|z−9√2−2i|2 is the distance of (9√2,2) from any point on circle.
Distance will be minimum when (9√2,2) will lie on the line joining the centre. PQ=OP−OQ
=9√2−2√2=7√2
PQ2=(7√2)2=98