Question

The value of such that $x^2-11x+a=0and{x}^2-14x+2a=0x$ may have a common root is

• A. 12
• B. 24
• C. 32

Answer 1

Consider the equations $x^2-11x+a=0$ ...(i) and $x^2-14x+2a=0x$ ...(ii) Let $\alpha ,\beta$ are the roots of equation (i) $\therefore$ Sum of roots of equation (i) is $\alpha +\beta =-\frac{B}{A}=\frac{(-11)}{1}=11$ ...(iii) Products of roots of equation (i) is $\alpha \beta =\frac{C}{A}=\frac{\alpha }{1}=\alpha$ ...(iv) Let $\alpha ,\gamma$ are the roots of equation (ii) $\therefore$Sum of roots of equation (ii) is $\alpha +\gamma =-\frac{B}{A}=-\frac{(-14)}{1}=14$ ...(v) Products of roots of equation (ii) is $\alpha \gamma =\frac{C}{A}=\frac{2\alpha }{1}=2\alpha$ ...(vi) Using equations, ()()()() solve for Thus we have, $\alpha =8,\beta =3,\gamma =6$ Substituting the values of $\alpha and\beta$ in equation (iv), we have $a=\alpha \beta \Rightarrow a=8\times 6=24$