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Find the Bi-linear Transformation which maps the points 1,i,-1 of z plane onto i,0,-i of w-plane
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written 2.8 years ago by |
$w=\dfrac{az+b}{cz+d}$............(i)
$i=\dfrac{a+b}{c+d}$, $0=\dfrac{ai+b}{ci+d}$, $-i=\dfrac{-a+b}{-c+d}$
ai-a=ci+c $\therefore c=a \dfrac{(i-1)}{(i+1)}$
$\therefore$ c = a.$\dfrac{(i-1).(i-1)}{(i+1).(i-1)}=a.\dfrac{(i^2-2i+1)}{i^2-1}=ai$
$\therefore $ c=d=ai
$\therefore$ w = $\dfrac{az-a}{aiz+ai}=\dfrac{z-1}{i(z+1)}=-i.\dfrac{(z-1)}{(z+1)}$
$|-i.\dfrac{(z-1)}{(z+1)}|=1$
$\therefore$ $|z-1|=|z+1|$.........$(|i|=1)$
$\therefore$ $|(x-1)-iy|=|(x+1)+iy|$
$\therefore$ (x-1)2 + y2 = (x+1)2 + y2
2x = -2x $\therefore$ 4x = 0 $\therefore$ x = 0