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  4. If θ denotes the acute angle between the curves, y = 10 - x2 and y = 2 + x2 at a point of their intersection, then | tan θ | is equal to :

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Q. If $\theta$ denotes the acute angle between the curves, $y = 10 - x^2$ and $y = 2 + x^2$ at a point of their intersection, then $| \tan \; \theta |$ is equal to :

JEE MainJEE Main 2019Application of Derivatives

Solution:

Point of intersection is $P(2,6)$.
Also, $m_{1} = \left(\frac{dy}{dx}\right)_{P\left(2,6\right)} = - 2x - 4 $
$ m_{2} =\left(\frac{dy}{dx}\right)_{P\left(2,6\right)} = 2x=4 $
$ \therefore \left|\tan\theta\right| = \left|\frac{m_{1} -m_{2}}{1+m_{1}m_{2}}\right| = \frac{8}{15} $