Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2007 / B / 7 / i / Solution

In the typical case, the line through $P,Q$, or their tangent when they coincide, is $v=mu+c$. Its intersections with $D$ have first coordinates given byTheir product is $c^2$. The third intersection is $-(P+Q)$, which has the same first coordinate as $P+Q$. ConsequentlyPassing to the quotient by rational squares gives $q(P)q(Q)q(P+Q)=1$. Every square class is its own inverse, soThe excluded cases have their specified special values; the source asks only for this chord-and-tangent argument away from them.