JEE MAIN - Mathematics (2011 - No. 5)
Let $$\alpha \,,\beta $$ be real and z be a complex number. If $${z^2} + \alpha z + \beta = 0$$ has two distinct roots on the line Re z = 1, then it is necessary that :
$$\beta \, \in ( - 1,0)$$
$$\left| {\beta \,} \right| = 1$$
$$\beta \, \in (1,\infty )$$
$$\beta \, \in (0,1)$$
Explanation
As real part of roots is $$1$$
Let roots are $$1 + pi,1 + q$$
$$\therefore$$ sum of roots $$ = 1 + pi + 1 + qi = - \alpha $$
which is real $$ \Rightarrow q = - p\,\,$$
or root are $$1+pi$$ and $$1-pi$$
product of roots $$ = 1 + {p^2} = \beta \in \left( {1,\infty } \right)$$
$$p \ne 0$$ as roots are distinct.
Let roots are $$1 + pi,1 + q$$
$$\therefore$$ sum of roots $$ = 1 + pi + 1 + qi = - \alpha $$
which is real $$ \Rightarrow q = - p\,\,$$
or root are $$1+pi$$ and $$1-pi$$
product of roots $$ = 1 + {p^2} = \beta \in \left( {1,\infty } \right)$$
$$p \ne 0$$ as roots are distinct.
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