It is well known that when given a null geodesic γ0(λ) with a point r in (p, q) conjugate to p along γ0(λ), there will be a variation of γ0(λ) which can give a time-like curve from p to q. Here we prove that the time-like curves coming from the above-mentioned variation (with the second derivative β2 ≠ 0) have a proper acceleration A = √AaAa which approaches infinity as the time-like curve approaches the null geodesic. Because the curve obtained from variation of the null geodesic must be everywhere time-like, we also discuss the constraint of the vector field Za on the null geodesic γ0(λ) cannot be zero.