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 which of the following is(are) correct?

• A. The value of a equals 6 .
• B. The common root is 4 .
• C. The value of a equals 8 .
• D. The common root is 2 .

Answer 1

Answer: (C), (D)

Now,
$\alpha +4\beta =6\text{ and }4\alpha \beta =a$
$\alpha +3\beta =c\text{ and }3\alpha \beta =6$
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 $\beta =1$.
If $\alpha =4$ and $4\beta =2$, then $\beta =\frac{1}{2}$
$\Rightarrow 3\beta =\frac{3}{2}$, which is non-integer.
$\therefore$ The common root is 2.