Problem 17P

17. (a) Use the identity for $\tan (x-y)$ (see Equation $14 \mathrm{~b}$ in Appendix $\mathrm{D}$ ) to show that if two lines $L_{1}$ and $L_{2}$ intersect at an angle $\alpha$, then

$$ \tan \alpha=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$

where $m_{1}$ and $m_{2}$ are the slopes of $L_{1}$ and $L_{2}$, respectively.

(b) The angle between the curves $C_{1}$ and $C_{2}$ at a point of intersection $P$ is defined to be the angle between the tangent lines to $C_{1}$ and $C_{2}$ at $P$ (if these tangent lines exist). Use part (a) to find, correct to the nearest degree, the angle between each pair of curves at each point of intersection.

(i) $y=x^{2} \quad$ and $\quad y=(x-2)^{2}$

(ii) $x^{2}-y^{2}=3$ and $x^{2}-4 x+y^{2}+3=0$

Step-by-Step Solution