Question

If $\sin \theta$ and $\cos \theta$ are the roots of the equation $a{x}^2-bx+c=0$, then which one of the following is correct?

• 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)

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

$a{x}^2-bx+c=0$
$\sin \theta$ and $\cos \theta$ are roots.
$\therefore \sin \theta +\cos \theta =\frac{b}{a};\sin \theta \cos \theta =\frac{c}{a}$
Then $(\sin \theta +\cos \theta )=\frac{b^2}{a^2}$
or ${\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =\frac{b^2}{a^2}$
or $1+2\frac{c}{a}=\frac{b^2}{a^2}$
or $\frac{a+2c}{a}=\frac{b^2}{a^2}$
or $\frac{a+2c}{a}\times a^2=b^2$
or $(a+2c)a=b^2$
or $a^2+2ac=b^2$
or $a^2-b^2+2ac=0$