# Angle of Intersection of Two Curves

Let y = f (x) and y = g (x) be two given intersecting curves. Angle of intersection of these curves is defined as the acute angle between the tangents that can be drawn to the given curves at the point of intersection.

Let (x1, y1) be the point of intersection

⇒ y1 = f (x1) = g (x1)

Slope of the tangent drawn to the curve y = f (x) at (x1, y1)

i.e. $\displaystyle m_1 = \frac{d f(x)}{dx}_{(x_1 , y_1)}$

Similarly slope of the tangent drawn to the curve y = g(x) at (x1, y1)

i.e. $\displaystyle m_2 = \frac{d g(x)}{dx}_{(x_1 , y_1)}$

If ‘ α ‘ be the angle (acute) of intersection, then

$\displaystyle tan\alpha = |\frac{m_1 – m_2}{1 + m_1 m_2}|$

If α = 0, then m1 = m2 .

Thus the given curves would touch each other at the point (x1, y1)

If α = π/2 , then m1 m2 = -1.

Thus the given curves would meet at right angle at the point (x1, y1) (or curves cut orthogonally at the point (x1, y1))