document.write( "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)
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Algebra.Com's Answer #796114 by greenestamps(13215) $\"\"$ $\"About$

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\n" ); document.write( "The latus rectum in this example is vertical, with length 6. The focus is, by definition, the midpoint of the latus rectum.

\n" ); document.write( "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.

\n" ); document.write( "Vertex form of the equation of a parabola opening left or right is

\n" ); document.write( " $\"x+=+%281%2F%284p%29%29%28y-k%29%5E2%2Bh\"$

\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "(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

\n" ); document.write( " $\"x+=+%281%2F6%29%28y%2B1%29%5E2-3.5\"$

\n" ); document.write( "(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

\n" ); document.write( " $\"x+=+%28-1%2F6%29%28y%2B1%29%5E2-0.5\"$

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