First slide

Theory of equations

Question

If the quadratic equation ax² + bx+ c = 0 (a > 0) has $\sec^{2} θ$ and ${cosec}^{2} θ$ as its roots, then which of the following must hold good?

Moderate

A

b + c = 0
B

$b^{2} - 4 ac \geq 0$
C

$c \geq 4 a$
D

$4 a + b \geq 0$

Solution

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

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