On quadratic families of CM elliptic curves

Given a CM elliptic curve with Weierstrass equation $y^2=f(x)$, and a positive definite binary quadratic form $Q(u,v)$, we show that there are infinitely many reduced integer pairs $(u,v)$ such that the twisted elliptic curve $Q(u,v)y^2=f(x)$ has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with $\vert u\vert, \vert v\vert \leq X$ is at least $X^{2-\varepsilon}$ for any $\varepsilon>0$.