Reduced equation of the vertical parabola

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Let's consider the vertical parabola, where the vertex is in the origin and the parabola lies on the $x$ axis.

The focus is now at point $F (0, \frac{p}{2})$ , and the equation of the generator line $D$ is: $y = - \frac{p}{2}$ .

The equation of the parabola is $x^{2} = 2 p y$

Example

Considering the equation $x^{2} = 12 y$ , find its focus, its generator line and its vertex.

The vertex is, by definition, at $A (0, 0)$ .

Comparing $x^{2} = 12 y$ with $x^{2} = 2 p y$ we can see that $2 p = 12$ and therefore $p = 6$ .

Substituting $p$ , we can find the focus $F (0, 3)$ and the generator line $y = - 3$ .