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If $\sin θ$ and $\cos θ$ are the roots of the equation $a x^{2} - b x + c = 0$ , then a correct statement is

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

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

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

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

answer is A.

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Detailed Solution

$\sin θ + \cos θ = \frac{b}{a}; \sin θ \cos θ = \frac{c}{a}$

$\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ = \frac{b^{2}}{a^{2}}$

$\Rightarrow 1 + \frac{2 c}{a} = \frac{b^{2}}{a^{2}} \Rightarrow \frac{b^{2}}{a^{2}} - \frac{2 c}{a} = 1$

$\Rightarrow \frac{b^{2} - 2 a c}{a^{2}} = 1$

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

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