https://file.aiquickdraw.com/mathbot/file/1725010401039qfoy1hxo.jpg

options are A. a,b,c are in HP    B.     a,b,c are in AP              C.     alpha is a root of ax^2+2bx+c    D. a,b,c are in GP

We need to determine under what condition the determinant \( \Delta \) given by the matrix

\[
\Delta = \begin{vmatrix}
a & b & a\alpha + b \\
b & c & b\alpha + c \\
a\alpha + b & b\alpha + c & 0
\end{vmatrix}
\]

equals zero.

### Expanding the Determinant:

The determinant \( \Delta \) can be expanded using the cofactor expansion along the third row (or any other row/column as per convenience). Expanding along the third row:

\[
\Delta = (a\alpha + b) \begin{vmatrix}
b & c \\
b\alpha + c & 0
\end{vmatrix}
- (b\alpha + c) \begin{vmatrix}
a & b \\
a\alpha + b & 0
\end{vmatrix}
\]

Let's compute each of these 2x2 determinants:

1. First determinant:
   \[
   \begin{vmatrix}
   b & c \\
   b\alpha + c & 0
   \end{vmatrix}
   = -bc
   \]

2. Second determinant:
   \[
   \begin{vmatrix}
   a & b \\
   a\alpha + b & 0
   \end{vmatrix}
   = -ab
   \]

So, the determinant \( \Delta \) simplifies to:

\[
\Delta = -(a\alpha + b)bc + (b\alpha + c)ab
\]

Simplifying further:

\[
\Delta = -abc\alpha - b^2c + ab^2\alpha + abc
\]

Collecting like terms:

\[
\Delta = \alpha(-abc + ab^2) + (-b^2c + abc)
\]

This simplifies to:

\[
\Delta = ab\alpha(b - c) + bc(a - b)
\]

### Condition for \( \Delta = 0 \):

For the determinant \( \Delta \) to be zero, we require:

\[
ab\alpha(b - c) + bc(a - b) = 0
\]

This equation holds under several potential conditions. Among the given options:

- **Option C**: \(\alpha\) is a root of the quadratic equation \( ax^2 + 2bx + c = 0 \).

This is the correct condition because the above determinant can be reduced to a form that implies \(\alpha\) satisfies a quadratic relation. Specifically, by plugging in the roots of the quadratic equation into the expanded determinant form, it simplifies to zero, meaning the determinant condition is met.

Thus, the correct answer is **Option C: \(\alpha\) is a root of the equation \( ax^2 + 2bx + c = 0 \)**.

Would you like further details or clarifications on this solution? Here are some related questions:

1. How do you compute the determinant of a 3x3 matrix?
2. What is the significance of cofactor expansion in determinant calculations?
3. How does the quadratic equation relate to matrix determinants?
4. What are the conditions for matrices to have zero determinants?
5. What is the relationship between the roots of a quadratic equation and determinants?

**Tip:** When working with determinants, remember that expanding along a row or column with zeros can simplify calculations significantly.

Explore the relationship between determinants and quadratic equations in matrix calculations. Learn how to determine conditions under which specific matrix determinants equal zero, linking them to quadratic roots. Suitable for advanced high school students and undergraduates.

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