Question

Find the value of '$a$' so that $x^2-11x+a=0$ and $x^2-14x+2a=0$ have a common root.

• A. $12$
• B. $24$
• C. $0$
• D. $48$

Answer 1

Answer: (B), (C)

$0$

$x^2-14x+2a=0$

$x^2-11x+a=0$

Let the common root be $\alpha$

$\frac{{\alpha }^2}{-14a+22a}=\frac{-\alpha }{a-2a}=\frac{1}{-11+14}$

$\Rightarrow \frac{{\alpha }^2}{8a}=\frac{\alpha }{a}=\frac{1}{3}$

$\Rightarrow \frac{\alpha }{a}=\frac{1}{3}$
$\Rightarrow \alpha =\frac{a}{3}$ ....... $\left(1\right)$

$\Rightarrow \frac{{\alpha }^2}{8a}=\frac{1}{3}$
$\Rightarrow {\alpha }^2=\frac{8a}{3}$

Using $\left(1\right)$
$\Rightarrow \frac{a^2}{9}=\frac{8a}{3}$
$\Rightarrow \frac{a^2}{9}-\frac{8a}{3}=0$
$\Rightarrow$ $a^2-24a=0$
$\Rightarrow a=0$ or $24$