If $a> 0$ and discriminant of $ax^2 +2bx + c $ is negative, then

$Δ=\begin{vmatrix}a & b & ax+b\\b & c & bx+c\\ax+b & bx+c & 0\end{vmatrix},$ is

negative

The correct answer is option (3) : negative

It is given that the discriminant of $ax^2 + 2bx + c $ is negative.

$∴4b^2 - 4ac < 0 ⇒b^2 -ac < 0 ⇒ac- b^2 > 0 $ ..............(i)

Also, $a> 0 $ and discriminant is negative.

$∴ax^2 + 2bx + c> 0 \, ∀ x \in R$ ...........(ii)

Now,

$Δ=\begin{vmatrix}a & b & ax+b\\b & c & bx+c\\ax+b & bx+c & 0\end{vmatrix}$

$⇒Δ=\begin{vmatrix}a & b & 0\\b & c & 0\\ax+b & bx+c & -x(ax+b)-(bx+c)\end{vmatrix}$

$⇒Δ=\begin{vmatrix}a & b & 0\\b & c & 0\\ax+b & bx+c & -(ax^2+2bx+c)\end{vmatrix}$

$⇒Δ= -(ax^2 +2bx + c) (ac-b^2 ) < 0 $    [Using (i) and (ii) ]