`z_(1) " and " z_(2)` are the roots of the equation `z^(2)-az+b=0, " where " abs(z_(1))=abs(z_(2))=1` and a, b are non-zero complex numbers, then

Options

A. `|a| le 1`
B. `|a| le 2`
C. `2arg(a) = arg(b)`
D. `agr a = 2arg(b)`

Answer

B, C

Explanation

`z_(1)` and `z_(2)` are the roots of the equation `z^(2) -az + b=0`. Hence, `z_(1)+z_(2) = a,z_(1)z_(2) = b`
Now `,|z_(1) +z_(2)| le |z_(1)| + |z_(2) |`
` rArr |z_(1) + z_(2)| = |a| le |1+1=2" "(because |z_(1)|=|z_(2)| =1)`
`rArr arg(a) = (1)/(2)[arg(z_(2) + arg(z_(1))]`
` (##CEN_ALG_C03_E13_027_S01.png" width="80%">
Also ,` arg(b) `= arg(z_(1)z_(2)) = arg(z_(1)) + arg(z_(2))`
`rArr 2 arg (a) = arg (b)`