Question

Let α, β, γ be the real roots of the equation, x³ + a x² + b x + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in u, v, w) given by α u + β v + γ w = 0; β u + γ v + α w = 0; γ u + α v + β w = 0 has non-trivial solution, then the value of a²/b is :

• A. 0
• B. 1
• C. 3
• D. 5

Hint:

The expression $\alpha^2+\beta^2+\gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha = 0$ implies $\alpha=\beta=\gamma$ for real numbers.

Answer 1

The Correct Option is C

To find the value of \(\frac{a^2}{b}\) given the conditions of the problem, we need to analyze both the cubic equation and the system of linear equations. The given cubic equation is:

\(x^3 + ax^2 + bx + c = 0\)

with roots \(\alpha, \beta, \gamma\). The system of equations is:

• \(\alpha u + \beta v + \gamma w = 0\)
• \(\beta u + \gamma v + \alpha w = 0\)
• \(\gamma u + \alpha v + \beta w = 0\)

For this system to have a non-trivial solution, the determinant of the system's coefficient matrix must be zero.

Consider the coefficient matrix:

\(\alpha\)	\(\beta\)	\(\gamma\)
\(\beta\)	\(\gamma\)	\(\alpha\)
\(\gamma\)	\(\alpha\)	\(\beta\)

The determinant of this matrix is:

\(\det = \alpha(\gamma^2 - \alpha\beta) - \beta(\beta^2 - \alpha\gamma) + \gamma(\beta\gamma - \alpha^2)\)

This simplifies to:

\(\det = \alpha\gamma^2 - \alpha^2\beta - \beta^3 + \alpha\beta\gamma + \beta\gamma^2 - \alpha^2\gamma\)

Given the nature of \(\alpha, \beta, \gamma\) being roots of the cubic equation, this determinant simplifies to zero if:

\(\alpha + \beta + \gamma = 0\) (using Vieta's formulas)

\(\alpha^2 + \beta^2 + \gamma^2 = 2(\alpha\beta + \beta\gamma + \gamma\alpha)\)

So, we have:

\(a = 0\)

As the question indicates that \(a, b ≠ 0\), to ensure a non-trivial solution and satisfy the determinant condition, we can consider:

\((\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)\)

This gives us:

\(a^2 = 3b\) 

Thus, the ratio \(\frac{a^2}{b} = 3\).

Therefore, the value of \(\frac{a^2}{b}\) is 3.