If the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0(\mathrm{a}>0)$ has ${\sec }^2 \mathrm{\theta }$and ${\mathrm{cosec}}^2 \mathrm{\theta }$as its roots, then which of the following must hold good?

${\sec }^2 \mathrm{\theta }+{\mathrm{cosec}}^2 \mathrm{\theta }={\sec }^2 \mathrm{\theta }\cdot {\mathrm{cosec}}^2 \mathrm{\theta }$
Sum of the roots is equal to their product and the roots are real. Hence,
$-\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{a}}$
or    $\mathrm{b}+\mathrm{c}=0$
Also,  ${\mathrm{b}}^2-4\mathrm{ac}\ge 0$
$$\begin{gathered} \begin{array}{l}\mathrm{or} {\mathrm{c}}^2-4\mathrm{ac}\ge 0\\ \mathrm{or} \mathrm{c}(\mathrm{c}-4\mathrm{a})\ge 0\\ \mathrm{or} \mathrm{c}-4\mathrm{a}\ge 0 (\because \mathrm{c}>0) \\ \mathrm{Further} \\ {\mathrm{b}}^2+4\mathrm{ab}\ge 0 \\ \mathrm{or} \mathrm{b}+4\mathrm{a}\le 0 (\because \mathrm{b}<0)\end{array} \end{gathered}$$