Angle of Intersection of Two Curves

Angle of Intersection of Two Curves

The angle of intersection of two curves is the angle subtended between the tangents at their point of intersection.

Let C₁ and C₂ be two curves having equations y = f(x) and y = g(x), respectively.

Let PT₁ and PT₂ be tangents to the curves C₁ and C₂ at their point of intersection.

Let θ be the angle between the two tangents PT₁ and PT₂ and θ₁ and θ₂ are the angles made by tangents with the positive direction of x-axis in anti-clockwise sense.

Then, \({{m}_{1}}=\tan {{\theta }_{1}}=\left( \frac{dy}{dx} \right){{C}_{1}}\) and \({{m}_{2}}=\tan {{\theta }_{2}}=\left( \frac{dy}{dx} \right){{C}_{2}}\).

From the figure it follows, θ = θ₂ – θ₁

\(\tan \theta =\tan \left( {{\theta }_{2}}-{{\theta }_{1}} \right)=\frac{\tan {{\theta }_{2}}-\tan {{\theta }_{1}}}{1+\tan {{\theta }_{2}}\tan {{\theta }_{1}}}\),

\(\tan \theta =\left| \frac{{{\left( \frac{dy}{dx} \right)}_{{{C}_{1}}}}-{{\left( \frac{dy}{dx} \right)}_{{{C}_{2}}}}}{1+{{\left( \frac{dy}{dx} \right)}_{{{C}_{1}}}}{{\left( \frac{dy}{dx} \right)}_{{{C}_{2}}}}} \right|=\left| \frac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}} \right|\).

Angle of intersection of these curves is defined as acute angle between the tangents.

Example: The acute angle between the curves y = |x² – 1| and y = |x² – 3| at their points of intersections is.

Solution: The points of intersection are (±√2, 1)

Since, the curves are symmetrical about y-axis, the angle of intersection at (-√2, 1)

= The angle of intersection at (√2, 1).

At (√2, 1); m₁ = 2x = 2√2; m₂ = – 2x = -2√2.

\(\tan \theta =\left| \frac{4\sqrt{2}}{1-8} \right|=\frac{4\sqrt{2}}{7}\),

\(\theta ={{\tan }^{-1}}\frac{4\sqrt{2}}{7}\).