Question 1091389
<br>The parabola has a y squared term, so it opens right or left.  The vertex form for a parabola that opens right or left is<br>
{{{(x-h) = (1/(4p))(y-k)^2}}}<br>
In this form, the vertex is (h,k), the length of the latus rectum is |4p|; and p is the distance from the vertex to the directrix and from the vertex to the focus.<br>
So put the equation for your parabola in that form:
{{{y^2 - 8y + 12x - 8 = 0}}}
{{{y^2 - 8y + 16 = -12x + 8 + 16}}}  (complete the square in y)
{{{(y-4)^2 = -12(x-2)}}}
{{{(x-2) = (-1/12)(y-4)^2}}}<br>
The vertex is (2,4).<br>
4p is -12, so p is -3, indicating that the parabola opens to the left ("in the negative direction").<br>
The directrix is a distance |p| to the right of the vertex; 2+3 = 5, so the directrix is the vertical line x=5.<br>
The focus is a distance |p| to the left of the vertex, at (-1,4).<br>
The length of the latus rectum is 12; since the focus is the midpoint of the latus rectum, the endpoints of the latus rectum are at (-1,10) and (-1,-2).<br>
{{{graph(300,200,-10,10,-10,20,sqrt(-12x+24)+4,-sqrt(-12x+24)+4)}}}