Question

The quadratic equations $x^2-6 x+a=0$ and $x^2-c x+6=0$ 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
$x ^2-6 x +8=0 \Rightarrow x =2,4$
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.