Question

The quadratic equations $x^2-6x+a=0,$ $x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$ . Then common root is :

• A. $2$
• B. $1$
• C. $4$
• D. $3$

Answer 1

The correct option is A $2$

Let $\alpha$ be the common root.

$x^2-6x+a=0\cdots \left(1\right)$
Let roots of above equation are $\alpha ,4\beta .$

$x^2-cx+6=0\cdots \left(2\right)$
Let roots of above equation are $\alpha ,3\beta .$

$\Rightarrow \alpha +4\beta =6,\alpha \left(4\beta \right)=a$
$\Rightarrow \alpha +3\beta =c,\alpha \left(3\beta \right)=6$
$\therefore a=8$

Hence, from equation $\left(1\right)$
$\Rightarrow x^2-6x+8=0\Rightarrow x=2,4$
$\therefore \alpha =2,\beta =1$