Question

Angle of Intersection of two Curves of Derivatives

Answer 1

At the point-of intersection of two curves, angle formed between their corresponding tangents is called angle of intersection of two curves.

Let y = f(x) and y = g(x) are two equations of curves. Let two curves intersect at point P.

Let PT₁ and PT₂ are two tangents at curves which interesect each other at point P and Φ is angle between them if tangents makes angle Ψ₁ and Ψ₂ with the x-axis then

m₁ = tanΨ₁ = slope of curve y = f(x) at point P

and m₂ = tanΨ₂ = slope of curve y = g(x) at point P

From figure, it is clear that

Φ = Ψ₁ - Ψ₂

∴ tan Φ = tan (Ψ₁ - Ψ₂)

Another angle between tangents is 180° - Φ. Generally smaller angle is taken as angle between tangents.

Orthogonal Curves: If angle of intersection of two curves is right angle or intersecting curves are perpendicular at their intersection are said to be orthogonal.

When curves will be orthogonal then

Φ = \(\frac{\pi}{2}\)

In this case, m₁ m₂ = - 1

⇒ \((\frac{dy}{dx}) y = f(x) = (\frac{dy}{dx}) _{y=g(x)} =-1\)