Question

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

• A. 1
• B. 4
• C. 3
• D. 2

Answer 1

Answer: (D)

Let $\alpha$ and $4\beta$ be roots of $x^2-6x+a=0$ and $\alpha ,3\beta$ be the roots of $x^2-cx+6=0$, then
$\alpha +4\beta =6$ and $4\alpha \beta =a$
$\alpha +3\beta =c$ and $3\alpha \beta =6$
We get $\alpha \beta =2$
$\Rightarrow a=8$
So the first equation is $x^2-6x+8=0$
$\Rightarrow x=2,4$
If $\alpha =2$ and $4\beta =4$ then $3\beta =3$
If $\alpha =4$ and $4\beta =2$, then $3\beta =3/2$
(non-integer)
$\therefore$ common root is $x=2$.