Let `x_(1),x_(2)` are the roots of the quadratic equation `x^(2) + ax + b=0`, where a,b, are complex numbers and `y_(1), y_(2)` are the roots of the quadratic equation `y^(2) + |a|yy+ |b| = 0`. If `|x_(1)| = |x_(2)|=1`, then prove that `|y_(1)| = |y_(2)| =1`

Explanation

`x^(2) + ax +b = 0` has roots `x_(1)` and `x_(2)` . Then `x_(1) + x_(2) = -a`
`x_(1) + x_(2) = -a`
`and x_(1)x_(2) = b`
From (2),
`|x_(1)| |x_(2)| = |b|`
`rArr |b|=1`
Also, `|-a| = |x_(1) +x_(2)|`
`rArr |a| le |x_(1)|+|x_(2)|`
`or |a| le 2`
Now `y^(2) + |a|y + |b| = 0` has roots `y_(1)` and `y_(2)` . Then
`y_(1),y_(2) =(-|a| pm sqrt(|a|^(2) -4|b|))/(2)`
`= (-|a| pm (sqrt(4-|a|^(2)))i)/(2)`
`rArr |y_(1)|,|y_(1)| = (sqrt(|a|^(2)+4-|a|^(2)))/(2) = 1`
Hence, `|y_(1)|=|y_(2)|= 1`