Question

If $a{x}^2+bx+1$ and $b{x}^2+ax+1$ have a common root, and $b\ne a$, then the common root is:

• A. $\frac{1}{a+b}$
• B. $\frac{1}{a-b}$
• C. $\frac{-1}{a+b}$
• D. none of these

Answer 1

The correct option is C $\frac{-1}{a+b}$

${ax}^2+bx+1=0{bx}^2+ax+1=0letcommonrootbe\alpha ,\alpha satisfiesboththeequations{a\alpha }^2+b\alpha +1=0-equation1{b\alpha }^2+a\alpha +1=0-equation2equation1-2(b^2\alpha +b)-({\alpha }^2\alpha +1)=0(b^2-a^2)\alpha +(b-a)=0\alpha =-\frac{b-a}{(b-a)(b+a)}\therefore \alpha =\frac{-1}{a+b}$