Question

Find all the tangents to the curve $y=\cos (x+y),-2\pi \le \times \le 2\pi$ that are parallel to the line $x+2y=0$

• A. $x+2y=\frac{\pi }{2}$ & $x+2y=-\frac{3\pi }{2}$
• B. $x+2y=\frac{\pi }{2}$ & $x+2y=-\frac{\pi }{2}$
• C. $x+2y=-\frac{3\pi }{2}$ & $x+2y=-\frac{3\pi }{2}$
• D. $x+2y=\frac{3\pi }{2}$ & $x+2y=-\frac{3\pi }{2}$
  

Answer 1

The correct option is A $x+2y=\frac{\pi }{2}$ & $x+2y=-\frac{3\pi }{2}$

Given equation is $x+2y=0$

Slope $=-\frac{1}{2}$
$=-\frac{1}{2}=-sin(x+y)(1+y^{\prime })$
$\Rightarrow -\frac{1}{2}=-\sin (x+y)(1-\frac{1}{2})$
$\Rightarrow \sin (x+y)=1$
$\Rightarrow \cos (x+y)=0\Rightarrow y=0\cos x=0$
$\therefore x=\frac{\pi }{2},\frac{3\pi }{2},-\frac{\pi }{2}.-\frac{3\pi }{2}$
$\sin x=1$ is possible for x = $\frac{\pi }{2}$ or $-\frac{3\pi }{2}$
Equation are : $y-0=-\frac{1}{2}(x-\frac{\pi }{2})$ and $y-0=-\frac{1}{2}(x+\frac{3\pi }{2})$