If sec `alpha and alpha` are the roots of `x^2-p x+q=0,` then (a) `p^2=q(q-2)` (b) `p^2=q(q+2)` (c)`p^2q^2=2q` (d) none of these

Explanation

(i) Let `z=|z|(costheta+isintheta)`
`implies|(z)/(|z|)-1|=|costheta+isintheta-1|`
`=sqrt((costheta-1)^(2)+sin^(2)theta)`
`=sqrt(2-2costheta)`
`=sqrt(4"sin"^(2)(theta)/(2))`
`=2|"sin"(theta)/(2)|`
`le2|(theta)/(2)|" "(because|sintheta|le|theta|)`
`=|theta|=|arg z|" "(1)`
(ii) `|z-1|=|z-|z|+|z|-1|`
`le|z-|z||+||z|-1|`
`=|z||(z)/(|z|)-1|+||z|-1|`
`le|z||arg z|+||z|-1|" "["From "(1)]`