Question 1138876
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The set of points equidistant from a fixed point (focus) and fixed line (directrix) is a parabola.<br>
With a vertical directrix at x=4 and a focus at (-4,-6.5), the parabola opens to the left.<br>
The vertex is the point on the axis of symmetry equidistant from the focus and directrix, so the vertex is (0,-6.5).<br>
The equation of a parabola opening left or right with vertex at (0,-6.5) is<br>
{{{(x-0) = (1/(4p))(y-(-6.5))^2}}}<br>
{{{x = (1/(4p))(y+6.5)^2}}}<br>
In the equation in that form, p is the directed distance from the vertex to the focus.  With the vertex at x=0 and the focus at x=-4, p = -4.<br>
So 4p = -16, and the completed equation is<br>
{{{x = (-1/16)(y+6.5)^2}}}