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Q.
The angle of intersection between the curves $y=\left[\left|\sin\,x\right|+\left|\cos\,x\right|\right]$ and $x^{2}+y^{2}=10, $ where $[x]$ denotes the greatest integer $≤ x,$ is
Given, $y=[|\sin x|+|\cos x|]$ and $x^{2}+y^{2}=10$
We know that $(|\sin x|+|\cos x|) \in[1, \sqrt{2}]$
$\therefore y=1$
The point of intersection of given curve is $x^{2}+1^{2}=10$
$\Rightarrow x^{2}=9$
$\Rightarrow x=\pm 3$
$\therefore $ Point of intersection is $(\pm 3,1)$
Now, $x^{2}+y^{2}=10$
$\Rightarrow 2 x+2 y \frac{d y}{d x}=0$
$\Rightarrow \frac{d y}{d x}=-\frac{x}{y}$
At point $(-3,1)$
$\frac{d y}{d x}=\frac{3}{1}=3$
$ \Rightarrow m_{1}=3$
Slope of line $y=1$ is $m_{2}=0$
$\therefore $ Angle between two curves is
$\tan \theta=\frac{m_{1}-m_{2}}{1+m_{1} \,m_{2}}=3$
$\Rightarrow \theta=\tan ^{-1}(3)$