Question

The quadratic equation $x^2-8x+a=0$ and $x^2-bx+16=0$ have one root in common and the other roots of $1^{\text{st }}$ and $2^{\text{nd }}$ equations are integers in the ratio $3:4$, then which of the following is(are) correct?

• A. The common root is a prime number
• B. The value of a is 12 .
• C. The value of b is 10 .
• D. The sum of $(a+b)$ is a rational number.

Answer 1

Answer: (A), (B), (C), (D)

$\alpha +3\beta =8\&3\alpha \beta =a$
$\alpha +4\beta =b\&4\alpha \beta =16\Rightarrow \alpha \beta =4\therefore a=12$
Hence, $x^2-8x+12=0$
$x=2\text{ or }x=6$
If common root is 2 then $\beta =2$ for which we get $3\beta$ and $4\beta$ both integers but if common root is 6 then $\beta =\frac{2}{3}$ for which $4\beta$ is not integer.
Hence, common root is 2