Question

If $\sin a$ and $\cos a$ are the roots of the equation $4x^2-kx-1=0,(k>0)$ then the value of $k$ is

• A. $2\sqrt{2}$
• B. $4$
• C. $2$
• D. $4\sqrt{2}$

Answer 1

The correct option is A $2\sqrt{2}$

Compare $4x^2-kx-1=0$ with quadratic equation $a{x}^2+bx+c=0$

Since, sum of roots $=\frac{-b}{a}=\frac{k}{4}$

product of roots $=\frac{c}{a}=\frac{-1}{4}$

$\therefore \sin \alpha +\cos \alpha =\frac{k}{4}$ .....(1)

and

$\sin \alpha \cos \alpha =\frac{-1}{4}$ .....(2)

By squaring equation (1) on both sides. we get,

$(\sin \alpha +\cos \alpha {)}^2=\frac{k^2}{16}$

$\Rightarrow {\sin }^2\alpha +{\cos }^2\alpha +2\sin \alpha \cos \alpha =\frac{k^2}{4}$

$\Rightarrow 1+2\sin \alpha \cos \alpha =\frac{k^2}{16}$ [$\because {\sin }^2\alpha +{\cos }^2\alpha =1$]

$\Rightarrow 1+2\left(\frac{-1}{4}\right)=\frac{k^2}{16}$ [using equation (2)]

$\Rightarrow \frac{1}{2}=\frac{k^2}{16}$

$\Rightarrow k^2=8$

$\Rightarrow k=2\sqrt{2}$

Option A is correct.