Question

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