Question

If $\sin \theta$ and $\cos \theta$ are the roots of ${ax}^2+bx+x=0$. Find the value of $a^2-b^2$

• A. 2ac
• B. -2ac
• C. 2ab
• D. -2bc

Answer 1

Answer: (B)

-2ac

Sum of roots $=\sin \theta +\cos \theta =\frac{-b}{a}$
Product of roots $=\frac{c}{a}$ i.e. $\sin \theta \times \cos \theta =\frac{c}{a}$

${(\sin \theta +\cos \theta )}^2={\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta$
$⟹{\left(\frac{-b}{a}\right)}^2=1+2\left(\frac{c}{a}\right)$
$⟹\frac{b^2}{a^2}=\frac{a+2c}{a}$

$⟹b^2=a^2+2ac$

$⟹a^2-b^2=-2ac$