Let $0 \neq a \in z$ and $A=\left[\begin{array}{ccc}a & a & a-y \\ a & a+x & a \\ a & a &…
Let $0 \neq a \in z$ and $A=\left[\begin{array}{ccc}a & a & a-y \\ a & a+x & a \\ a & a & a\end{array}\right]$ be a matrix. Then, equation det $A=16$ represents
a parabola
a circle
an ellipse
a rectangular hyperbola
Solution
We have,
$\operatorname{det}(A)=\left|\begin{array}{ccc}a & a & a-y \\ a & a+x & a \\ a & a & a\end{array}\right|$
Apply $C_1 \rightarrow C_1-C_3$
$\operatorname{det}(A)=\left|\begin{array}{ccc}y & a & a-y \\ 0 & a+x & a \\ 0 & a & a\end{array}\right|$
$\Rightarrow \quad|A|=y\left(a^2+a x-a^2\right)$
$\Rightarrow \quad|A|=a x y$
$\Rightarrow \quad 16=$ axy $\quad$ [given $\operatorname{det} A=16$ ]
$\Rightarrow \quad x y=\frac{a}{16}$
which represent a rectangular hyperbola.