equation of parabola having its focus at S(2,0) and one extrimity of its latus ractum as (2,2) is ?

Question

equation of parabola having its focus at S(2,0) and one extrimity of its latus ractum as (2,2) is ?

Answer 1

There is not a unique answer. There are two answers.

The general form of a parabola with center (h,k) that opens sideways as this one does is:

4a(x - h) = (y - k)²

We know the parabola opens sideways because the latus rectum is vertical. It is parallel to the directrix and runs from one side of the parabola thru the focus to the other side. In this case, by symmetry, that is from (2,2) to (2,-2).

The length of the latus rectum is:

4a = 4
a = 1

The vertex is a distance of a = 1 from the focus. That puts the vertex at either (1,0) or (3,0).

So the equation of the parabola is:

4a(x - h) = (y - k)²

4(x - 1) = (y - 0)²
or
4(x - 3) = (y - 0)²

Simplifying:

4(x - 1) = y²
or
4(x - 3) = y²