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Let $p$ and $q$ be two distinct odd primes, $p<q$ and $E_{p,q}:y^2=x^3-pqx$be an elliptic curve. Fix a line $L_{a.b}:y=\frac{a}{b}x$ where $a\in\mathbb{Z},b\in \mathbb{N}$ and $(a,b)=1$. We study sufficient conditions that$p$ and $q$ must satisfy so that there are infinitely many elliptic curves$E_{p,q}$ that intersect $L_{a,b}$.

On families of elliptic curves $E_{p,q}:y^2=x^3-pqx$ that intersect the same line $L_{a,b}:y=\frac{a}{b}x$ of rational slope