Multiple choice

The determinant $\left| {\begin{array}{*{20}{c}}a&b&{a\alpha + b}\b&c&{b\alpha + c}\{a\alpha + b}&{b\alpha + c}&0\end{array}} \right|$ is equal to zero, if-

  1. $a, b, c$ are in AP
  2. $a, b, c$ are in GP
  3. $\alpha $ is a root of the equation $a{x^2} + bx + c = 0$
  4. $\left( {x - \alpha } \right)$ is a factor of $a{x^2} + 2bx + c$
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D Correct answer
Explanation

The determinant represents the condition for the quadratic ax^2 + 2bx + c = 0 to have a root alpha. Expanding the determinant or using properties of determinants shows that it equals -(a*alpha^2 + 2b*alpha + c)^2. For this to be zero, a*alpha^2 + 2b*alpha + c must be zero, meaning (x-alpha) is a factor of ax^2 + 2bx + c.

AI explanation

Expanding the given determinant along the third row yields the equation (a*alpha + b)^2 * c - 2(a*alpha + b)(b*alpha + c)b + (b*alpha + c)^2 * a = 0. Factoring out (a*alpha + b), we can rewrite this relationship as -(a*alpha + b) * (a(alpha)^2 + 2b*alpha + c) = 0. This reveals that either a*alpha + b = 0 or alpha is a root of the equation ax^2 + 2bx + c = 0. The latter condition means that (x - alpha) is a factor of the quadratic expression ax^2 + 2bx + c.

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