Equation of the vertical parabola with generic vertex

Let's consider the vertical parabola with the vertex at a generic point $A (x_{0}, y_{0})$ .

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The focus is on $F (x_{0}, y_{0} + \frac{p}{2})$ and the generator line is given by the equation $y = y_{0} - \frac{p}{2}$ .

The equation of the parabola is $(x - x_{0})^{2} = 2 p (y - y_{0})$

Example

Given the parabola $x^{2} - 8 y + 16 = 0$ , find its focus, its vertex and the equation of its generator line.

First we should express the equation of the parabola in the form $(x - x_{0})^{2} = 2 p (y - y_{0})$ .

To do this we can add $8 y - 16$ on both sides, and take $8$ as common factor: $x^{2} = 8 (y - 2)$

Expressed as $(x - 0)^{2} = 2 \cdot 4 (y - 2)$ we obtain all the necessary information.

Then we can identify $x_{0} = 0, y_{0} = 2, p = 4$ .

The focus is on $F (x_{0}, y_{0} + \frac{p}{2})$ , in this case $F (0, 4)$ .

The vertex is on $A (x_{0}, y_{0})$ i.e. $A (0, 2)$ .

The equation of the generator line is $y = y_{0} - \frac{p}{2}$ , in our case is $y = 0$ .