Question

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

Answer 1

The vertex form of a parabola of this type is:

`y = 1/(4f)(x-h)^2+k" [1]"`

where `(h,k)` is the vertex and `f = y_"focus"-k`.

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

`h = (4+ (-2))/2 = 1`

`y = 1/(4f)(x-1)^2+k" [2]"`

We know that `4f` is `+-` the length of the latus rectum:

`4f = 4 - (-2)` or `4f = -2 -4`

`4f = 6` or `4f = -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 = 1/6(x-1)^2+k" [3]"`

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

`-7 = 1/6(4-1)^2+k`

`-7 = 1/2+k`

`k = -7.5`

`y = 1/6(x-1)^2-7.5 larr` answer.