Question
If the equations $${x^2} + ix + a = 0,{x^2} - 2x + ia = 0,a \ne 0$$ have a common root then
A.
$$a$$ is real
B.
$$a = \frac{1}{2} + i$$
C.
$$a = \frac{1}{2} - i$$
D.
the other root is also common
Answer :
$$a = \frac{1}{2} - i$$
Solution :
As all the coefficients are not real, one common root does not imply that the other root is also common.
Let $$\alpha $$ be the common root. Then $${\alpha ^2} + i\alpha + a = 0,{\alpha ^2} - 2\alpha + ia = 0$$
$$\eqalign{
& \Rightarrow \,\,\frac{{{\alpha ^2}}}{a} = \frac{\alpha }{{a\left( {1 - i} \right)}} = \frac{1}{{ - 2 - i}} \cr
& \Rightarrow \,\,{a^2}{\left( {1 - i} \right)^2} = a\left( { - 2 - i} \right). \cr} $$