A similar question has been asked and answered on Mathoverflow:
It is a "classical" fact that any nonsingular plane cubic curve $C$
can be projectively generated by means of a pencil of lines and a
pencil of conics.
The starting point of the construction is the observation that the
lines defined by any $g_2^1$ on $C$ all pass through the same point
$p$ of $C$, that following Sylvester is called the coresidual point.
Now, take four points $q_1, \ldots, q_4$ on $C$, such that any three
of them are not collinear. Let $p \in C$ be the coresidual point with
respect to the $g_2^1$ on $C$ cut by the pencils of conics through
$q_1, \ldots, q_4$. Therefore such a pencil of conics and the pencil
of lines through $p$ projectively generate $C$.
All of this is explained in the paper by N. Fraser Kötter's
synthetic geometry of algebraic curves, Proceedings of the
Edinburgh Mathematical Society 7, 46–61 (1888). However, the
language is rather old-fashioned so the article is not easy to read
A modern treatment can be found in Dolgachev's book Classical
Algebraic Geometry, Section 3.3 (this is the googlebook link).
See also Bunch, On the Straight Line Construction of Unicursal Cubics. Here the construction uses a 1-2 correspondence between two pencils of lines.
And for something more down to earth: Matheq, A General Construction for Circular Cubics