# How to Find Angle Between Two Curves - Formula, Example

## Find Angle Between Two Curves at Point of Intersection - Tutorial

#### Definition

When two curves intersect each other the angle at the intersecting point is called as angle of intersection between two curves.

#### Formula

tan(θ) = (m2-m1)/(1+(m1.m2)) ∀ m2>m1 tan(θ) = (m1-m2)/(1+(m1.m2)) ∀ m1>m2
###### Where,
m1 = Curve 1 Tangent line slope m2 = Curve 2 Tangent line slope

#### Example

Find the acute angle between the two curves y=2x2 and y=x2-4x+4

##### Given ,

Here the 2 curves are represented in the equation format as shown below y=2x2 --> (1) y=x2-4x+4 --> (2) Let us learn how to find angle of intersection between these curves using this equation.

##### Solution :
###### Step 1 :

Solving equ 1 and equ 2 2x2 = y x2- 4x + 4 = y x2+ 4x - 4 = 0 By factorizing the quadratic equation x2+ 4x - 4 = 0 we get the x values as x = 0.8 and x = -4.8 From the x values the maximum value (0.8) is substituted in equation 1 to find y value

###### Where,

y = 2x(0.8)2 y = 1.3 From this values we get (0.8,1.3), which is an intersect point of curve.

###### Step 2 :

Differentiate equ.1 and equ.2 Differentiation of equ 1 y=2x2 dy/dx = 4x --> (3)

###### Where,

dy/dx(x2) = 2x Differentiation of equ 2 y=x2-4x+4 dy/dx = 2x - 4 --> (4)

###### Where ,

dy/dx(x) = 1 and dy/dx(constant) = 0

###### Step 3 :

Find the slope by substituting intersect point (0.8,1.3) in equ.3 and equ.4, Equ. 3 4x = 4(0.8) = 3.2 = m1 Equ. 4 2x - 4 = 2(0.8) - 4 = -2.4 = m2

###### Step 4 :

Find the Angle by substituting slope values in Formula tan(θ) = (m1-m2)/(1+(m1.m2)) ∀ m1>m2 From formula θ = tan-1[(m1-m2)/(1+(m1.m2))] θ = tan-1((3.2+2.4)/(1+(3.2*-2.4)) θ = tan-1(5.6/-6.68) θ = tan-1(0.8383) θ = 39.974 ° Therefore, the angle of intersection between the given curve is θ = 39.974 ° 