Question

The equation of one of the tangents to the curve $y=\cos (x+y),-2\pi \le x\le 2\pi$ that is parallel to the line $x+2y=0$, is

• A. $x+2y=1$
• B. $x+2y=\pi /2$
• C. $x+2y=\pi /4$
• D. None of these

Answer 1

Answer: (B)

$y=\cos (x+y)\dots$(i)
$\therefore \frac{dy}{dx}=-\sin (x+y)\left\{1+\frac{dy}{dx}\right\}$
$\therefore \frac{dy}{dx}=-\frac{\sin (x+y)}{1+\sin (x+y)}=-\frac{1}{2}$
$\Rightarrow \sin (x+y)=1$, so $\cos (x+y)=0$
$\therefore$ from (i), $y=0$ and $(x+y)=2n\pi +\frac{\pi }{2}$
Tangent at $\left(\frac{\pi }{2},0\right)$ is $x+2y=\frac{\pi }{2}$