Choose a point $$P(x_0,0)$$ in the $$x$$-axis. Find the equation of the parabola whose focus coincides with point $$P$$ and the origin with the vertex. Find its generator line.
See development and solution
Development:
We choose $$P(1,0)$$.
First, the origin is the vertex, so $$A(0,0)$$ and it is a reduced equation. The point $$P(1,0)$$ is the focus $$F(\dfrac{p}{2},0)$$, so $$\dfrac{p}{2}=1$$ and then $$p=2$$.
It is possible now to find the equation by substituting $$p$$ in $$y^2=2px$$. The equation is $$$y^2=4x$$$
To obtain the generator function we substitute $$p$$ in $$x=-\dfrac{p}{2}$$ and find the straight line $$$x=-1$$$
Solution:
For $$P(1,0)$$ the parabola is $$y^2=4x$$ and the generator line is $$x=-1$$.