On the intersections of rational curves with cubic plane curves

Let C be a smooth cubic curve in the complex projective plane. We show that for every positive integer k, there are only finite number of rational curves of degree k each intersects the cubic C at exactly one point. The number of such rational curves is known when k is 1, as a smooth cubic has 9 flexes points. This number seems to be closely related to the number of plane rational curves of degree k passing through 3k-1 general points, which has been computed.