Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 29 / 2 / ii / Solution

At $P_1=(0,0)$ the tangent slope is $-2$, so its equation is $y=-2x$. Substitution leaves $x^3=0$; this tangent meets the curve three times at $P_1$. The chord-and-tangent group law therefore givesIn particular $P_1$ has exact order three. The chord through $P_1$ and $P_2=(-2,-4)$ is $y=2x$. Substitution gives $x(x+2)(x-2)=0$, so the third intersection is $(2,4)$. Negating it gives