FINDING THE ANGLE BETWEEN TWO CURVES

Problem 1 :

Find the angle between the rectangular hyperbola xy = 2 and the parabola x² + 4y = 0.

Solution :

To find the angle between two curves, we have to find slope of the tangent line drawn to the curve at the point of intersection of these two curves.

(1) = (2)

2/x = -x²/4

8 = -x³

x³ = -8

x³ = (-2)³

x = -2

Applying x = -2 in (1), we get

y = 2/(-2) ==> -1

So, the point of intersection is (-2, -1).

xy = 2

x(dy/dx) + y (1) = 0

x(dy/dx) = -y

dy/dx = -y/x

m₁ = -y/x

x² + 4y = 0

2x + 4(dy/dx) = 0

4(dy/dx) = -2x

dy/dx = -2x/4

dy/dx = -x/2

m₂ = -x/2

So, the angle between two curves is 45 degree.

Problem 2 :

Find the angle between the curves y = x² and x = y² at their points of intersection (0, 0) and (1, 1).

Solution :

y = x² and x = y²

Angle between the tangents at (1, 1) :

Angle between the tangents at (0, 0) :