First slide

Theory of equations

Question

If $\sin α, \cos α$ are the roots of the equation $a x^{2}$ + $b x + c = 0,$ $(a \neq 0)$ ,then

Moderate

A

$a^{2} - b^{2} + 2 a c = 0$
B

$a^{2} + b^{2} + 2 a c = 0$
C

$(a - c)^{2} = b^{2} + c^{2}$
D

none of these

Solution

$\sin α + \cos α = - \frac{b}{a}, \sin α \cos α = \frac{c}{a}$
Now $1 = \sin^{2} α + \cos^{2} α$
$= (\sin α + \cos α)^{2} - 2 \sin α \cos α$
$\Rightarrow 1 = \frac{b^{2}}{a^{2}} - \frac{2 c}{a} \Rightarrow a^{2} = b^{2} - 2 a c$
$\Rightarrow a^{2} - b^{2} + 2 a c = 0$

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