 Solution

Nonsingularity with $b=a^2$ forces $a\ne 0$. TakeIt lies on the curve. Its tangent slope is $a/(2a)=1/2$, so the doubling formula gives $x(2P)=1/4$. If $P$ were torsion, $2P$ would be torsion as well, contradicting the integrality just proved. Therefore $P$ has infinite order.