Question

If the equations $x^2-ax+b=0$ and $x^2+bx-a=0$ have a common root, then select the correct statement(s):

• A. $a-b=1$
• B. $a+b=2$
• C. $a-b=3$
• D. $a+b=0$

Answer 1

Answer: (A), (D)

$a+b=0$

Let $\alpha$ be a common root of $x^2-ax+b=0$ and $x^2+bx-a=0$

$\therefore {\alpha }^2-a\alpha +b=0\cdots \left(i\right)$
and ${\alpha }^2+b\alpha -a=0\cdots \left(ii\right)$

Subtracting both $\left(i\right)\&\left(ii\right),$ we get:

$({\alpha }^2-a\alpha +b)-({\alpha }^2+b\alpha -a)=0$

$\Rightarrow (a+b)-(a+b)\alpha =0$

$\Rightarrow (a+b)(1-\alpha )=0$

$\therefore a+b=0$ and $\alpha =1$

On putting $\alpha$ in equation ${\alpha }^2-a\alpha +b=0$, we get

$1^2-a.1+b=0\Rightarrow a-b=1$

Hence, $a+b=0$ and $a-b=1$