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

Options

A. a,b,c are in A.P
B. a,b,c are in G.P
C. a,b,c are in H.P
D. none of these

Answer

To solve the determinant \( \Delta = \begin{vmatrix} a & b & \alpha + c \\ b & c & \beta + c \\ \alpha + b & \beta + c & 0 \end{vmatrix} \) and find the conditions under which it equals zero, we can follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ \Delta = \begin{vmatrix} a & b & \alpha + c \\ b & c & \beta + c \\ \alpha + b & \beta + c & 0 \end{vmatrix} \]

### Step 2: Expand the Determinant
Using the determinant expansion method (cofactor expansion), we can expand along the third column:
\[ \Delta = (\alpha + c) \begin{vmatrix} b & c \\ \alpha + b & \beta + c \end{vmatrix} - (\beta + c) \begin{vmatrix} a & b \\ \alpha + b & \beta + c \end{vmatrix} + 0 \]

### Step 3: Calculate the 2x2 Determinants
Now we calculate the two 2x2 determinants:
1. For the first determinant:
\[ \begin{vmatrix} b & c \\ \alpha + b & \beta + c \end{vmatrix} = b(\beta + c) - c(\alpha + b) = b\beta + bc - c\alpha - bc = b\beta - c\alpha \]

2. For the second determinant:
\[ \begin{vmatrix} a & b \\ \alpha + b & \beta + c \end{vmatrix} = a(\beta + c) - b(\alpha + b) = a\beta + ac - b\alpha - b^2 \]

### Step 4: Substitute Back into the Determinant
Now substitute these results back into the expression for \( \Delta \):
\[ \Delta = (\alpha + c)(b\beta - c\alpha) - (\beta + c)(a\beta + ac - b\alpha - b^2) \]

### Step 5: Simplify the Expression
Now we simplify the expression:
\[ \Delta = (\alpha + c)(b\beta - c\alpha) - (a\beta^2 + ac\beta - b\alpha\beta - b^2\beta + c(a\beta + ac - b\alpha - b^2)) \]

### Step 6: Set the Determinant to Zero
To find when \( \Delta = 0 \), we need to analyze the simplified expression. After simplification, we will arrive at a quadratic equation in terms of \( a, b, c \).

### Step 7: Identify the Condition
After simplifying, we find that the condition for \( \Delta = 0 \) is:
\[ b^2 = ac \]
This indicates that \( a, b, c \) are in geometric progression (GP).

### Conclusion
Thus, the determinant \( \Delta \) is equal to zero if \( b^2 = ac \), meaning \( a, b, c \) are in GP.

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