Q. $E_{1}:a + b + c = 0$, if $l$ is a root of $ax^{2} + bx + c = 0,\ E_{2}:b^{2} - a^{2} = 2ac$, if $\sin{\theta,\cos\theta}$ are the roots of $ax^{2} + bx + c = 0$. Which of the following is true?
A) $E_{1}$ is true, $E_{2}$ is true
B) $E_{1}$ is true, $E_{2}$ is false
C) $E_{1}$ is false $E_{2}$ is true
D) $E_{1}$ is false, $E_{2}$ is false
Explanation
(a)
Since, 1 is root of $ax^{2} + bx + c = 0$
$\Rightarrow \ \ \ a + b + c = 0$
$\therefore E_{1}\ :a + b + c = 0$ is true
Since, $\cos{\theta,\sin\theta}$ are the roots of $ax^{2} + bx + c = 0$
$\therefore\ \ \ \sin{\theta + \cos{\theta = - \frac{b}{a}}}$
And $\sin{\theta\cos{\theta = \frac{c}{a}}}$
$\Rightarrow \ \ \ \ \ \left( \sin{\theta + \cos\theta} \right)^{2} = \frac{b^{2}}{a^{2}}$
$\Rightarrow \ \sin^{2}{\theta + \cos^{2}{\theta + 2\sin{\theta\cos{\theta = \frac{b^{2}}{a^{2}}}}}}$
$\Rightarrow \ \ \ 1 + 2\left( \frac{c}{a} \right) = \frac{b^{2}}{a^{2}}$
$\Rightarrow b^{2} - a^{2} = 2ac$
Hence, $E_{1}$ and $E_{2}$ both are true
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.