Question

If $\sin\,\theta$ and $\cos \theta $ are the roots of the equation $ax^2 - bx + c = 0$, then $a, b$ and $c$ satisfy the relation

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

Answer 1

Answer: (B)

Since, $\sin \theta$ and $\cos \theta$ are the roots of the equation $a x^{2}-b x+c=0$
$\therefore \sin \theta+\cos \theta=\frac{b}{a}$ and $\sin \theta \cos \theta=\frac{c}{a}$
$\Rightarrow \left(\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta\right)=\frac{b^{2}}{a^{2}}$
$\Rightarrow 1+2 \sin \theta \cos \theta=\frac{b^{2}}{a^{2}}$
$\Rightarrow 1+2 \times \frac{c}{a}=\frac{b^{2}}{a^{2}}$
$\Rightarrow a^{2}-b^{2}+2 a c=0$