Question

The acute angle of intersection of the curves $y=[\left|\sin x\right|+\left|\cos x\right|]$ and $x^2+y^2=5$ (where $[.]$ denotes the greatest integer function) is ${\tan }^{-1}\left(k\right)$ then $k$ is

Answer 1

The value of $\left|\sin x\right|+\left|\cos x\right|$ will lie between
$1\le \left|\sin x\right|+\left|\cos x\right|\le \sqrt{2}$
$\Rightarrow [\left|\sin x\right|+\left|\cos x\right|]=1$
$\Rightarrow y=1$
Other curve is $x^2+y^2=5;$
Putting $y=1$ we get $x=\pm 2$
$\Rightarrow$ Points of intersection are $(2,1)$ and $(-2,1)$

$\because$ the curves intersect each other symmetrically, thus angle of intersection will be same at both the points
$\Rightarrow$Slope of $y=1$ will be $0$
$\Rightarrow m_1=0$
$x^2+y^2=5$
Differentiating it w.r.t. $x$
$\Rightarrow$ $2x+2y\frac{dy}{dx}=0$
$\Rightarrow m_2=-\frac{x}{y}$
Slope $m_2$ at $(-2,1)$ is $2$
Angle between two Curves is equal to the angle between the tangents drawn at their point of intersection.
Angle between two lines is given by formula
$\theta ={\tan }^{-1}\left|\frac{m_2-m_1}{1+m_1\cdot m_2}\right|$
Putting values of $m_1,m_2$
$\Rightarrow \theta ={\tan }^{-1}\left(2\right)$
$\Rightarrow k=2$