Chapter 3, Problem 21P

(a) Use the identity for tan(x – y) (see Equation 14b in Appendix D) to show that if two lines L₁, and L₂ intersect at an angle α, then

$\tan \alpha =\frac{m_2-m_1}{1+m_1{m}_2}$

where m₁, and m₂ are the slopes of L₁ and L₂ respectively.

(b) The angle between the curves C₁ and C₂ at a point of intersection P is defined to be the angle between the tangent lines to C₁, and C₂ at P (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair or curves at each point of intersection.

(i) y = x² and y = (x – 2)²

(ii) x² – y² = 3 and x² – 4x + y² + 3 = 0