 Solution

Write $x=c/d$ in lowest terms and put $H=\max \{|c|,|d|\}\ge 1$. For $P\ne O,(0,0)$,Before reduction to lowest terms, both numerator and denominator are bounded in magnitude by $(1+|a|+|b|)H^2$. Cancellation can only decrease the height. Hence one may chooseThe duplication formula for $C$ isThe numerator has magnitude at most $(1+|b|{)}^2{H}^4$, and the denominator at most $4(1+|a|+|b|)H^4$. Thussatisfies the second bound. At the identity or any point whose relevant image is the identity, the image height is zero and the same nonnegative constants work directly. Thus the formulas cover every rational point, including the exceptional points of the displayed coordinate expressions.