Question

If $\Delta =\left|\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right|;0\le \theta <2\pi$ then $\Delta \in [a,b]$ Find $\frac{b}{a}?$

Answer 1

$\Delta =\left|\begin{array}{ccc}1& sin\theta & 1\\ -sin\theta & 1& sin\theta \\ -1& -sin\theta & 1\end{array}\right|$
$\Rightarrow \Delta =(1+{\sin }^2\theta )-\sin \theta (-\sin \theta +\sin \theta )+({\sin }^2\theta +1)$
$\Rightarrow \Delta =2+2{\sin }^2\theta$
${\sin }^2\theta \in [0,1]$
$\Rightarrow \Delta \in [2,4]$
$\Rightarrow a=2$ and $b=4$
$\therefore \frac{b}{a}=2$