Find the equation of parabola whose latus rectum is the line segment joining the points (-3, 1), (1, 1)

You need to remember the equation of parabola, such that:

`4p(x-h)= (y-k)^2`

The length of latus rectum is |4p| and it must be set equal to the length of the segment joining the points (-3,1), (1,1).

You need to find the length of the segment joining the points (-3,1), (1,1), using the distance formula, such that:

`d = sqrt((-3-1)^2 + (1-1)^2)`

`d = sqrt 16`

`d = 4`

Hence, the length of latus rectum is:

`|4p| = 4 => p = +-1`

The focus of parabola is the midpoint of the segment line joining the points (-3,1), (1,1), hence, you may evaluate the coordinates of the midpoint, such that:

`x = (-3+1)/2 => x = -1`

`y = (1+1)/2 => y = 1`

The coordinates of the focus are (-1,1).

The vertex of parabola is `(-1+-p, 1)` , hence, the coordinates of the vertex are `(-1+-1, 1).`

The parabola that opens to the right has the length of latus rectum `4p = 4` , the vertex `(h,k) = (-1-1, 1) = (-2,1) ` and the equation of parabola `4(x + 2) = (y -1)^2.`

The parabola that opens to the left has the length of latus rectum `4p = -4` , the vertex `(h,k) = (-1+1, 1) = (0,1)` and the equation of parabola `-4(x-0) = (y -1)^2.`