Given that `b^2 - ac lt 0,a lt 0` then the value of
`|(a,b,ax+by),(b,c,bx+cy),(ax+by,bx+cy,0)|`

Options

A. 0
B. `+ive`
C. `-ive`
D. `b^2+ac`

Answer

To solve the determinant \( D = \begin{vmatrix} a & b & ax + by \\ b & c & bx + cy \\ ax + by & bx + cy & 0 \end{vmatrix} \) given the conditions \( b^2 - ac < 0 \) and \( a < 0 \), we can follow these steps:

### Step 1: Apply Row Operations
We can simplify the determinant using row operations. Let's perform the operation \( R_3 \leftarrow R_3 - R_1 - R_2 \).

This gives us:
\[ D = \begin{vmatrix} a & b & ax + by \\ b & c & bx + cy \\ 0 & 0 & 0 \end{vmatrix} \]

### Step 2: Expand the Determinant
Notice that the third row is now all zeros. The determinant of a matrix with a row of zeros is zero. Therefore, we can conclude:
\[ D = 0 \]

### Step 3: Verify the Conditions
Given the conditions \( b^2 - ac < 0 \) and \( a < 0 \), we can analyze the implications:
- The condition \( b^2 - ac < 0 \) indicates that the quadratic polynomial formed by the second row and the third row has no real roots, which implies it does not cross the x-axis.
- The condition \( a < 0 \) suggests that the parabola opens downwards.

### Conclusion
Since the determinant simplifies to zero, we conclude that:
\[ \text{The value of the determinant } D = 0. \]