SOLUTION: Please help me with this question :- Find the equation of the parabola whose focus is (-2,-1) and the latus rectum joins the points (-2,2) and (-2,-4)

Question 1171190: Please help me with this question :- Find the equation of the parabola whose focus is (-2,-1) and the latus rectum joins the points (-2,2) and (-2,-4)

Answer by greenestamps(13214) (Show Source): You can put this solution on YOUR website!

The latus rectum in this example is vertical, with length 6. The focus is, by definition, the midpoint of the latus rectum.

So the parabola opens EITHER right or left -- with the only given information being the endpoints of the latus rectum, we don't know which. So there will be two answers to the problem.

Vertex form of the equation of a parabola opening left or right is

where (h,k) is the vertex and p is the directed distance from the vertex to the focus. In this form of the equation, |4p| is the length of the latus rectum.

So we know |4p|=6, which means p is either 1.5 or -1.5. That means the vertex is 1.5 units either right or left of the focus.

(1) p=1.5: parabola opens to the right; focus is 1.5 units to the right of the vertex, which means the vertex is 1.5 units to the left of the focus -- at (-3.5,-1). Then the equation is

(2) p=-1.5: parabola opens to the left; focus is 1.5 units to the left of the vertex, which means the vertex is 1.5 units to the right of the focus -- at (-0.5,-1). Then the equation is

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