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. $(a-c{)}^2=b^2+c^2$
• C. $a^2+b^2+2ac=0$
• D. $a^2-b^2+2ac=0$

Answer 1

The correct option is D

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

Given: $\sin \alpha ,\cos \alpha$ are the roots of the equation $a{x}^2+bx+c=0$,
Using relation between roots and coefficients,
$\sin \alpha +\cos \alpha =-\frac{b}{a},\sin \alpha .\cos \alpha =\frac{c}{a}$

Using the trigonometric identity,
$1={\sin }^2\alpha +{\cos }^2\alpha 1=(\sin \alpha +\cos \alpha {)}^2-2\sin \alpha .\cos \alpha 1=\frac{b^2}{a^2}-\frac{2c}{a}\Rightarrow a^2-b^2+2ac=0$