Question

If an equation of a tangent to the curve, $y=\cos \left(x+y\right),−1≤x≤1+π$ , is $x+2y=k$ then $k$ is equal to

• A. $1$
• B. $2$
• C. $\frac{Ï€}{4}$
• D. $\frac{Ï€}{2}$

Answer 1

Answer: (D)

Let $y=\cos x+y$
$⇒\frac{dy}{dx}=−\sin x+y1+\frac{dy}{dx}….1$
Now, given equation of tangent is
$x+2y=k$
$⇒$ Slope $=\frac{−1}{2}$
So, $\frac{dy}{dx}=\frac{−1}{2}$ put this value in 1 , we get
$\frac{−1}{2}=−\sin x+y1−\frac{1}{2}$
$⇒\sin x+y=1$
$⇒x+y=\frac{π}{2}⇒y=\frac{π}{2}−x$
Now, $\frac{π}{2}−x=\cos x+y$
$⇒x=\frac{π}{2}$ and $y=0$
Thus $x+2y=k⇒\frac{π}{2}=k$