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  4. The acute angle between the curves x y=2 and y2=4 x at their point of intersection is

Q. The acute angle between the curves $x y=2$ and $y^{2}=4 x$ at their point of intersection is

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Solution:

$xy = 2 ....$ (1)
and $y^2 = 4x ....$(2)
$x\frac{dy}{dx} + y = 0$
$2y \frac{dy}{dx} = 4$
solving (1) and (2) we get $x =1$ and $y =2$
therefore $m _{1}=-2 \,\,\,m _{2}=1$
$\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right|=\left|\frac{-2-1}{1+(-2)(1)}\right|=3 $
The angle is $\tan ^{-1} 3$

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