Question

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

• A. $b+c=0$
• B. $b^2-4ac\ge 0$
• C. $c\ge 4a$
• D. $4a+b\ge 0$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $b+c=0$
B $b^2-4ac\ge 0$
C $c\ge 4a$

${\sec }^2\theta$ & ${\csc }^2\theta$ are the roots of the equation $a{x}^2+bx+c=0$
Since the roots are real.
$\therefore b^2-4ac\ge 0$
Sum of the roots of the equation $={\sec }^2\theta +{\csc }^2\theta =\frac{-b}{a}$
$\Rightarrow \frac{{\sin }^2\theta +{\cos }^2\theta }{{\sin }^2\theta {\cos }^2\theta }=\frac{-b}{a}$
$\frac{1}{{\sin }^2\theta {\cos }^2\theta }=\frac{-b}{a}$ ...(1)
Product of the roots of the equation $={\sec }^2\theta {\csc }^2\theta =\frac{c}{a}$
$\Rightarrow \frac{1}{{\sin }^2\theta {\cos }^2\theta }=\frac{c}{a}$ ...(2)
$\Rightarrow {\sin }^{2}2\theta =\frac{4a}{c}\le 1\Rightarrow c\ge 4a$
Combining (1) & (2) we get,
$b+c=0$
Hence, options 'A', 'B' and 'C' are correct.