Question 1171190
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The latus rectum in this example is vertical, with length 6.  The focus is, by definition, the midpoint of the latus rectum.<br>
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.<br>
Vertex form of the equation of a parabola opening left or right is<br>
{{{x = (1/(4p))(y-k)^2+h}}}<br>
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.<br>
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.<br>
(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<br>
{{{x = (1/6)(y+1)^2-3.5}}}<br>
(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<br>
{{{x = (-1/6)(y+1)^2-0.5}}}<br>