Question

The determinant$\left| \begin{matrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{matrix} \right| = 0$ if

• A. a, b, c are in A.P.
• B. a, b, c are in G.P. or $(x - \alpha)$ is a factor of $ax^{2} + 2bx + c = 0$
• C. a, b, c are in H.P.
• D. $\alpha$ is a root of the equation

Answer 1

Answer: (B)

a, b, c are in G.P. or $(x - \alpha)$ is a factor of $ax^{2} + 2bx + c = 0$

Answer 2

Applying $R_{3} \rightarrow R_{3} - \alpha R_{1} - R_{2}$, we get

a & b & a\alpha + b \\ b & c & b\alpha + c \\ 0 & 0 & - a\alpha^{2} - b\alpha - b\alpha - c \end{matrix} \right| = 0$$ $$\Rightarrow - (a\alpha^{2} + 2b\alpha + c)(ac - b^{2}) = 0 \Rightarrow a\alpha^{2} + 2b\alpha + c = 0$$ or $b^{2} = ac$ $\Rightarrow$ $x = \alpha$ is a root of $ax^{2} + 2bx + c = 0$ or a, b, c are in G.P. $\Rightarrow (x - \alpha)$ is a factor of $ax^{2} + 2bx + c = 0$ or a, b, c are in G.P.