If roots of the equation $z^2 + \alpha z + \beta = 0$ lie on $|z| = 1$, then
- $2|Im \space \alpha| = 1 - |\beta|^2$
- $2|Im \space \alpha| = |\beta|^2 - 1$
- $Im \space \alpha = 0$
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None of these
If roots z1, z2 lie on |z|=1, then z1*z2 = beta and z1+z2 = -alpha. Since |z1|=|z2|=1, |beta| = |z1*z2| = 1. Also, z2 = 1/conj(z1). Then -alpha = z1 + 1/conj(z1). This implies alpha is not necessarily real, but the relationship between alpha and beta is complex. Given the options, 'None of these' is the standard conclusion for this specific property.
Let the roots of z squared plus alpha z plus beta equals 0 be z1 and z2. Since both roots lie on the unit circle (absolute value of z equals 1), we have the magnitude of z1 equal to 1 and the magnitude of z2 equal to 1. By Vieta's formulas, z1 plus z2 equals negative alpha and z1 multiplied by z2 equals beta. Taking the magnitude of the product, the magnitude of beta equals the magnitude of z1 multiplied by the magnitude of z2, which equals 1 times 1, so the magnitude of beta is 1. The options provided do not satisfy this universal condition for all possible roots on the unit circle, meaning none of the constraints given in options A, B, or C are generally true. The result is None of these.