# How do you find an equation of a parabola given endpoints of latus rectum are (-2,-7) and (4,-7)?

Aug 3, 2017

The vertex form of a parabola of this type is:

$y = \frac{1}{4 f} {\left(x - h\right)}^{2} + k \text{ }$

where $\left(h , k\right)$ is the vertex and $f = {y}_{\text{focus}} - k$.

The x coordinate of the vertex, h, is the midpoint between the x coordinates of the two points:

$h = \frac{4 + \left(- 2\right)}{2} = 1$

$y = \frac{1}{4 f} {\left(x - 1\right)}^{2} + k \text{ }$

We know that $4 f$ is $\pm$ the length of the latus rectum:

$4 f = 4 - \left(- 2\right)$ or $4 f = - 2 - 4$

$4 f = 6$ or $4 f = - 6$

We are not told whether the parabola opens up or down and you have only asked for one of the two possible equations, therefore, I shall choose the positive value:

$y = \frac{1}{6} {\left(x - 1\right)}^{2} + k \text{ }$

We can find the value of k substituting in one of the points:

$- 7 = \frac{1}{6} {\left(4 - 1\right)}^{2} + k$

$- 7 = \frac{1}{2} + k$

$k = - 7.5$

$y = \frac{1}{6} {\left(x - 1\right)}^{2} - 7.5 \leftarrow$ answer.