Angle of Intersection of Two Curves

Here you will learn angle of intersection of two curves formula with examples.

Let’s begin –

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.

Let \(C_1\) and \(C_2\) be two curves having equations y = f(x) and y = g(x) respectively.

and \(m_1\) = slope of tangent to y = f(x) at P = \(({dy\over dx})_{C_1}\)

and \(m_2\) = slope of the tangent to y = g(x) at P = \(({dy\over dx})_{C_2}\)

Angle between the curve is \(tan \phi\) = \(m_1 – m_2\over 1 + m_1 m_2\)

Orthogonal Curves 

If the angle of two curves is at right angle, the two curves are equal to intersect orthogonally and the curves are called orthogonal curves.

If the curves are orthogonal then \(\phi\) = \(\pi\over 2\)

\(\therefore\) \(m_1 m_2\) = -1

Note : Two curves \(ax^2 + by^2\) = 1 and \(a’x^2 + b’y^2\) = 1 will intersect orthogonally, if

\(1\over a\) – \(1\over b\) = \(1\over a’\) – \(1\over b’\)

Example : find the angle between the curves xy = 6 and \(x^2 y\) =12.

Solution : The equation of the two curves are

xy = 6          …….(i)

and, \(x^2 y\) = 12            …………(ii)

from (i) , we obtain y = \(6\over x\). Putting this value of y in (ii), we obtain

\(x^2\) \((6\over x)\) = 12 \(\implies\) 6x = 12 

\(\implies\) x = 2

Putting x = 2 in (i)  or (ii), we get y = 3. 

Thus, the two curves intersect at P(2, 3).

Differentiating (i) with respect to x, we get

x\(dy\over dx\) + y = 0 \(\implies\) \(dy\over dx\) = \(-y\over x\)

\(\implies\) \(m_1\) = \(({dy\over dx})_{(2, 3)}\) = \(-3\over 2\)

Differentiating (ii) with respect to x, we get

\(x^2\) \(dy\over dx\) + 2xy  = 0 \(\implies\) \(dy\over dx\) = \(-2y\over x\)

\(\implies\) \(m_2\) = \(({dy\over dx})_{(2, 3)}\) = -3

Let \(\theta\) be the angle, then

\(tan \theta\) = \(m_1 – m_2\over 1 + m_1 m_2\) = \(3\over 11\)

\(\theta\) = \(tan^{-1} (3/11)\)

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