Question

f $b^2-ac<0,a<0$ then the value of
$\left|\begin{array}{ccc}a& b& ax+by\\ b& c& bx+cy\\ ax+by& bx+cy& 0\end{array}\right|$ is

• A. Zero
• B. Positive
• C. Negative
• D. $b^2+ac$

Answer 1

Answer: (B)

Let $\Delta =\left|\begin{array}{ccc}a& b& ax+by\\ b& c& bx+cy\\ ax+by& bx+cy& 0\end{array}\right|$
$=\left|\begin{array}{ccc}a& b& ax+by\\ b& c& bx+cy\\ 0& 0& -\left(a{x}^2+2bxy+c{y}^2\right)\end{array}\right|$
[Applying $R_3\to R_3-x{R}_1-y{R}_2$]
$=\left(b^2-ac\right)\left(a{x}^2+2bxy+c{y}^2\right)$
Now, $b^2-ac<0$ and $a<0$
$\Rightarrow$ Discriminant of $a{x}^2+2bxy+c{y}^2$ is negative and $a<0$.
$\Rightarrow a{x}^2+2bxy+c{y}^2<0$ for all $x,y\in R$ [See Quadratics]
$\Rightarrow \Delta =\left(b^2-ac\right)\left(a{x}^2+2bxy+c{y}^2\right)>0.$