Need urgent help with a complex locus problem! (1 Viewer)

[emilios]

http://imgur.com/KQKDl7F

Squares have too many properties... I have no idea which one to use. I tried some stuff with the diagonals being equal but that led to nowhere. All I've gathered is that P=2cis(theta) and Q = 1/2cis(-theta). Also, if P = x+iy then x^2+y^2 = 4, and Q = 1/4(x-iy)

 

[jyu]

http://imgur.com/KQKDl7F

Squares have too many properties... I have no idea which one to use. I tried some stuff with the diagonals being equal but that led to nowhere. All I've gathered is that P=2cis(theta) and Q = 1/2cis(-theta). Also, if P = x+iy then x^2+y^2 = 4, and Q = 1/4(x-iy)

Midpoint of PQ is (5x+3yi)/8
Translate P to P': x+yi - (5x+3yi)/8 = (3x+5yi)/8
Rotate P' by 90 degree anticlockwise about O to get Q': i(3x+5yi)/8 = (-5y+3xi)/8
Translate Q' to Q: (-5y+3xi)/8 + (5x+3yi)/8 = [5(x-y)+3(x+y)]/8
Let X=5(x-y)/8 and Y=3(x+y)/8; 8X/5=x-y and 8Y/3=x+y; 64X^2/25+64Y^2/9=2(x^2+y^2)=8
.: X^2/(25/8)+Y^2/(9/8)=1
a=sqrt(25/8) and b=sqrt(9/8)