Let [], [], [] be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c [] R and…

Question

Let $$\alpha$$, $$\beta$$, $$\gamma$$ be the real roots of the equation, x3 + ax2 + bx + c = 0, (a, b, c $$\in$$ R and a, b $$\ne$$ 0). If the system of equations (in u, v, w) given by $$\alpha$$u + $$\beta$$v + $$\gamma$$w = 0, $$\beta$$u + $$\gamma$$v + $$\alpha$$w = 0; $$\gamma$$u + $$\alpha$$v + $$\beta$$w = 0 has non-trivial solution, then the value of $${{{a^2}} \over b}$$ is

Accepted Answer

x3 + ax2 + bx + c = 0

Roots are $$\alpha$$, $$\beta$$, $$\gamma$$.

For non-trivial solutions,

$$\left| {\begin{pmatrix} \alpha  & \beta  & \gamma   \
   \beta  & \gamma  & \alpha   \
   \gamma  & \alpha  & \beta   \

\end{pmatrix} } \right| = 0$$

$$ \Rightarrow $$ $${\alpha ^3} + {\beta ^3} + {\gamma ^3} - 3\alpha \beta \gamma  = 0$$

$$ \Rightarrow $$ $$(\alpha  + \beta  + \gamma )\left[ {{{\left( {\alpha  + \beta  + \alpha } \right)}^2} - 3\left( {\sum {\alpha \beta } } \right)} \right] = 0$$

$$ \Rightarrow $$ $$( - a)[{a^2} - 3b] = 0$$

$$ \Rightarrow $$ ${a^2} = 3b$ ($$ \because $$ a $$ \ne $$ 0)

$$ \Rightarrow {{{a^2}} \over b} = 3$$

Solutions · 1