Question

A value of $b$ for which the equations $x^2+bx-1=0,x^2+x+b=0$ have one root in common is

• A. $-\sqrt{2}$
• B. $i\sqrt{5}$
• C. $\sqrt{3}i$
• D. $\sqrt{2}$

Answer 1

Answer: (C)

$\sqrt{3}i$

$\begin{array}{c}Given\\ x^2+bx-1=0\\ and,x^2+x+b=0haveonecommonroots\\ Letcommonrootsbe\alpha \\ \Rightarrow {\alpha }^2+b\alpha -1=0and{\alpha }^2+\alpha +b=0\\ \Rightarrow \frac{{\alpha }^2}{-1-b^2}=\frac{-\alpha }{-1-b}=\frac{1}{b-1}\\ \Rightarrow \alpha =\frac{b+1}{1+b},\alpha =-\frac{1+b^2}{1+b}and{\alpha }^2=\frac{b^2+1}{1-b}\\ \Rightarrow {\left(1+b\right)}^2=\left(1+b^2\right)\left(1-b\right)\\ \Rightarrow b^2=-3orb=\sqrt{3}i.\end{array}$

Hence,option $C$ is the correct answer.