Let `a ,b ,a n dc` be any three nonzero complex number. If `|z|=1a n d' z '` satisfies the equation `a z^2+b z+c=0,` prove that `a a =c c a n d|a||b|=sqrt(a c( b )^2)`

Explanation

If `|z|=1`
` rArr .barz = 1 or barz = (1)/(2)`
We have , `az^(2) + bz + c=0`
Taking conjugate on both sides, we get
`barz barz^(2) + barb barz + barc =0`
Replacing `barz` by `(1)/(z)` in (2), we get
`barc(z)^(2) + barbz + bara =0`
Now, (1) and (3) must be indentical equaitons,
`rArr (a)/(barc) = (b)/(barc) = (c)/(bara)`
or `a.bara = c.barc`
and `bara b= cbar`
From `veca. b = c. barb`, we get
`abara b barb = a(barb)^(2) c`
`rArr |a||b| = sqrt(ac(bar b)^(2)) and |a|=|c|`