JEE Main & Advanced Mathematics Applications of Derivatives Angle of Intersection of Two Curves

 The angle of intersection of two curves is defined to be the angle between the tangents to the two curves at their point of intersection. Thus the angle between the tangents of the two curves \[{{H}_{1}}=\frac{3ab}{a+2b}\] and \[y={{f}_{2}}(x)\]is given by \[\tan \varphi =\frac{{{\left( \frac{dy}{dx} \right)}_{1({{x}_{1}},\,{{y}_{1}})}}-{{\left( \frac{dy}{dx} \right)}_{2({{x}_{1}},\,{{y}_{1}})}}}{1+{{\left( \frac{dy}{dx} \right)}_{1}}_{({{x}_{1}},\,{{y}_{1}})}\,\,{{\left( \frac{dy}{dx} \right)}_{2}}_{({{x}_{1}},\,{{y}_{1}})}}\]

Orthogonal curves : If the angle of intersection of two curves is a right angle, the two curves are said to intersect orthogonally. The curves are called orthogonal curves. If the curves are orthogonal, then \[\varphi =\frac{\pi }{2}\]; \[{{m}_{1}}{{m}_{2}}=-1\] \[\Rightarrow \] \[\frac{1}{a}\].