the determinant `|{:(a,,b,,aalpha+b),(b,,c,,balpha+c),(aalpha+b,,balpha+c,,0):}|=0` is equal to zero if

Options

A. a,b,c are in A.P
B. a,b,c are in G.P.
C. `alpha` is a root of the equation `ax^(2) +bx+c=0`
D. `(x-alpha)` is a factor fo `ax^(2) +2bx+c`

Explanation

To solve the determinant equation

\[ \left| \begin{array}{ccc} a & b & \alpha + b \\ b & c & \beta + c \\ \alpha + b & \beta + c & 0 \end{array} \right| = 0, \]

we will follow these steps:

### Step 1: Write the determinant
We start with the determinant:

\[ \Delta = \left| \begin{array}{ccc} a & b & \alpha + b \\ b & c & \beta + c \\ \alpha + b & \beta + c & 0 \end{array} \right|. \]

### Step 2: Apply row operations
We can simplify the determinant by performing row operations. We can subtract the first column multiplied by \( \alpha \) from the third column:

\[ \Delta = \left| \begin{array}{ccc} a & b & b \\ b & c & c \\ \alpha + b & \beta + c & -\alpha \end{array} \right|. \]

### Step 3: Expand the determinant
Now, we will expand the determinant using the third column:

\[ \Delta = a \left| \begin{array}{cc} b & c \\ \beta + c & -\alpha \end{array} \right| - b \left| \begin{array}{cc} b & c \\ \alpha + b & -\alpha \end{array} \right| + (-\alpha) \left| \begin{array}{cc} b & c \\ \alpha + b & \beta + c \end{array} \right|. \]

### Step 4: Calculate the 2x2 determinants
Calculating the 2x2 determinants, we have:

1. For the first determinant:
\[ \left| \begin{array}{cc} b & c \\ \beta + c & -\alpha \end{array} \right| = b(-\alpha) - c(\beta + c) = -b\alpha - c\beta - c^2. \]

2. For the second determinant:
\[ \left| \begin{array}{cc} b & c \\ \alpha + b & -\alpha \end{array} \right| = b(-\alpha) - c(\alpha + b) = -b\alpha - c\alpha - bc. \]

3. For the third determinant:
\[ \left| \begin{array}{cc} b & c \\ \alpha + b & \beta + c \end{array} \right| = b(\beta + c) - c(\alpha + b) = b\beta + bc - c\alpha - bc = b\beta - c\alpha. \]

### Step 5: Substitute back into the determinant
Substituting these back into our determinant expression gives us:

\[ \Delta = a(-b\alpha - c\beta - c^2) - b(-b\alpha - c\alpha - bc) - \alpha(b\beta - c\alpha). \]

### Step 6: Set the determinant to zero
Setting \(\Delta = 0\) leads to a polynomial equation in terms of \(\alpha\):

\[ -a b \alpha - a c \beta - a c^2 + b^2 \alpha + b c \alpha + b^2 c - \alpha b \beta + \alpha c \alpha = 0. \]

### Step 7: Factor the polynomial
Factoring out common terms and simplifying will lead to conditions on \(a\), \(b\), and \(c\).

### Conclusion
The determinant equals zero if either:

1. \( a \alpha^2 + 2b \alpha + c = 0 \) (which indicates that \(\alpha\) is a root of this polynomial).
2. \( a c = b^2 \) (which indicates that \(a\), \(b\), and \(c\) are in geometric progression).