Question

Find the acute angles between the curves at their points of intersection. y = x², y = x³

Answer 1

The angle between the curves is same as the angle between their tangents at the points of intersection. We find the points of intersection of

y = x² … (1)

and y = x³ … (2)

From (1) and (2)

x³ = x²

x³ = x² = 0

x²(x - 1) = 0

∴ x = 0 or x = 1 

When x = 0, y = 0. 

When x = 1, y = 1.

∴ equation of tangent to y = x at P is y = 0. 

∴ the tangents to both curves at (0, 0) are y = 0

∴ angle between them is 0. 

Angle at P = (1, 1) 

Slope of tangent to y = x² at P

∴ equation of tangent to y = x³ at P is y – 1 = 3(x – 1) y = 3x – 2 

We have to find angle between y = 2x – 1 and y = 3x – 2 

Lines through origin parallel to these tagents are y = 2x and y = 3x 

∴ and These lines lie in XY-plane. 

∴ the direction ratios of these lines are 1, 2, 0 and 1, 3, 0.

The angle θ between them is given by

Hence, the required angles are 0 and \(cos^{-1} \left(\frac{7}{5\sqrt{2}} \right).\)