Question

# If $\alpha ,\beta$are the roots of the quadratic equation $a{x^2} + bx + c = 0$ and $3{b^2} = 16ac$ then:A.$\alpha = 4\beta {\text{ or }}\beta = 4\alpha$B.$\alpha = - 4\beta {\text{ or }}\beta = - \alpha$C.$\alpha = 3\beta {\text{ or }}\beta = 3\alpha$D.$\alpha = - 3\beta {\text{ or }}\beta = - 3\alpha$

Hint: Use the information that roots always satisfy the quadratic equation. Also, if quadratic equation is in the standard from, which is $l{x^2} + mx + n = 0$then sum of the roots $= \dfrac{{ - m}}{l}$and multiplication of the roots $= \dfrac{n}{l}$.
According to the given information in the hint, $\alpha + \beta = \dfrac{{ - b}}{a}$ and $\alpha \beta = \dfrac{c}{a}$. Then ${(\alpha + \beta )^2} = \dfrac{{{b^2}}}{{{a^2}}}$ . we have given, $3{b^2} = 16ac$. On putting the value of ${b^2}$ we’ll get,
${(\alpha + \beta )^2} = \dfrac{{{b^2}}}{{{a^2}}} \Rightarrow {b^2} = {a^2}{(\alpha + \beta )^2} \\ 3{b^2} = 16ac \\ \Rightarrow 3{a^2}{(\alpha + \beta )^2} = 16a \times a(\alpha \beta ){\text{ }}[{\text{Using, }}\alpha \beta = \dfrac{c}{a}] \\ \Rightarrow 3{a^2}({\alpha ^2} + {\beta ^2} + 2\alpha \beta ) = 16{a^2}\alpha \beta \\ \Rightarrow 3({\alpha ^2} + {\beta ^2} + 2\alpha \beta ) = 16\alpha \beta \\ \Rightarrow 3{\alpha ^2} + 3{\beta ^2} + 6\alpha \beta = 16\alpha \beta \\ \Rightarrow 3{\alpha ^2} - 10\alpha \beta + 3{\beta ^2} = 0 \\ \Rightarrow 3{\alpha ^2} - (9\beta + \beta )\alpha + 3{\beta ^2} = 0 \\ \Rightarrow 3{\alpha ^2} - 9\beta \alpha - \beta \alpha + 3{\beta ^2} = 0 \\ \Rightarrow 3\alpha (\alpha - 3\beta ) - \beta (\alpha - 3\beta ) = 0 \\ \Rightarrow (\alpha - 3\beta )(3\alpha - \beta ) = 0 \\ \Rightarrow \alpha = 3\beta ,\dfrac{\beta }{3} \\$
Hence, $\alpha = 3\beta$ and $\beta = 3\alpha$ are the correct set of equations. So, option C will be the correct option.