The determinant $$\begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \end{vmatrix}$$ is equal to zero, if:
Step-by-Step Solution
Key Concept: The determinant equals zero when either $ac - b^2 = 0$ (condition for G.P: $b^2 = ac$) or $a + 2b + c = 0$ (which makes $(x-a)$ a factor of $ax^2 + 2bx + c$ since $a(a)^2 + 2b(a) + c = 0$). These conditions arise from factoring the determinant after row operations.
The determinant $\begin{vmatrix} a & b & aa+b \\ b & c & ba+c \\ aa+b & ba+c & 0 \end{vmatrix} = 0$ is evaluated using row operations. After performing $R_3 \to R_3 - aR_1 - R_2$, the third row becomes $[0, 0, -(aa+b)-ba-c]$, leading to the condition $-(aa^2 + 2ba + c)(ac - b^2) = 0$. Therefore, either $ac - b^2 = 0$ or $aa^2 + 2ba + c = 0$, meaning $a, b, c$ are in G.P. or $a$ is a root of $ax^2 + 2bx + c = 0$.
Correct Answer: 2,4