Equation of the horizontal parabola with generic vertex

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

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In this case the focus is on $F (x_{0} + \frac{p}{2}, y_{0})$ and the generator line is $x = x_{0} - \frac{p}{2}$ .

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

Example

The equation of the parabola which focus point is at $F (- 2, 4)$ and the vertex point at $A (3, 4)$ .

Identify $A (x_{0}, y_{0})$ with $A (3, 4)$ and $F (x_{0} + \frac{p}{2}, y_{0})$ with $F (- 2, 4)$ . We obtain $x_{0} = 3$ and $y_{0} = 4$ .

By analyzing the focus and the generic equation we know that $x_{0} + \frac{p}{2} = 3 + \frac{p}{2} = - 2$ , then $\frac{p}{2} = 5$ and we obtain the parameter value $p = 10$ .

Substituting into the equation $(y - y_{0})^{2} = 2 p (x - x_{0})$ we obtain $(y - 4)^{2} = 20 (x - 3)$