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. $4$
• B. $3$
• C. $2$
• D. $1$

Answer 1

The correct option is C $2$

Let the roots of the first and second equations are $(\alpha ,4\beta )$ and $(\alpha ,3\beta )$ respectively

$\therefore$ product of the roots are $4\alpha \beta =a$ ....(1)

and $3\alpha \beta =6\Rightarrow \alpha \beta =2$

$\Rightarrow a=8$ . ....From eqn (1)

Therefore, the roots of the equation $x^2-6x+8=0$ are $(2,4)$.

Given that the other roots of the first and second equations are integers, so if $\alpha =4\Rightarrow \beta =\frac{1}{2}$

In this case $3\beta$ is not integer.

Therefore, common root will be $\alpha =2$