Question

If sin $\alpha$ & cos $\alpha$ are the roots of the equation $a{x}^2+bx+c=0$, then

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

Answer 1

The correct option is A $a^2-b^2+2ac=0$

$\sin \alpha$ and $\cos \alpha$ are the roots of $a{x}^2+bx+c=0$

$\Rightarrow \sin \alpha +\cos \alpha =\frac{-b}{a}$ and $\sin \alpha \cos \alpha =\frac{c}{a}$

$(\sin \alpha +\cos \alpha {)}^2={\sin }^2\alpha +{\cos }^2\alpha +2\sin \alpha \cos \alpha$

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

$\Rightarrow b^2=a^2+2ac$

$\therefore a^2-b^2+2ac=0$

Hence, option A.