Two curves touch each other if the angle between the tangents to the curves at the point of intersection is 0o, in which case we will have,tanΨ1= tanΨ2.

Hello sandeep !

First of all i would franky say that i didn't understand what to find or prove in the question ...

Angle between curves can be simply calculated by differentiating the function of curve ...!

Suppose two curves are there with equation f1(x) and f2(x)

df1(x)/dx will give the slope of tangent at f1(x)  and df2(x)/dx will give the slope of tangent at f2(x)

Suppose angle of tangent at f1(x) is  A and at f2(x) is B ..

then tan(A) - tan(B) = df1(x)/dx - df2(x)/dx

tan(A-B).[1 + tanA.tanB] = df1(x)/dx - df2(x)/dx   ( tan(A-B) = (tanA - tanB)/(1+tanA.tanB)

Hence if the angle between the curve is zero ...that is ...A-B = 0

So ...tan(A) = tan(B)    

Regards

Yagya

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