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Angle between two curves

In order to measure the angle between two curves, we measure the angle between the tangents to the curves at that point. And that is obtained by the formula below:

tan θ = delim{|}{{m_1~-~m_2}/{1~+~m_1m_2}}{|}

where θ is the angle between the 2 curves, and m1 and m2 are slopes or gradients of the tangents to the curve at the point of intersection.

 

Let’s look at this through an example below:

Find the acute angle between the curves y = x2 and y = (x – 3)2

The curve intersects at the point (where the two equations are equal). So

x2 = (x – 3)2

x2 =  x2 – 6x + 9

6x – 9 = 0

x = 9/6  =   3/2

y = x2 = ({3/2})^2  = {9/4}

The intersection point is ({3/2},~{9/4})

The tangent to y = x2 at ({3/2},~{9/4}) has a gradient of

dy/dx = 2x  =  2*{3/2}  =  3  = m1

The tangent to y = (x – 3)2 at ({3/2},~{9/4}) has a gradient of

dy/dx = 2(x – 3)1 =  2.({3/2}~-~3)  =  -2.({3/2})  =  -3  = m2

tan θ = delim{|}{{m_1~-~m_2}/{1~+~m_1m_2}}{|}

= delim{|}{{3~-~(-3)}/{1~+~-3*3}}{|}

= delim{|}{{3~+~3}/{1~-~3*3}}{|}

= delim{|}{{6}/{1~-~9}}{|}

= delim{|}{{-6}/{8}}{|}

= 6/8  =  3/4

tan θ = 3/4

θ = tan^-1{(3/4)}

=  36° 52′ (to the nearest minute)