A parabola can be drawn given a focus of (−5,7) and a directrix of x=−3. What can be said about the parabola?
The vertex of the parabola is halfway between the focus and directrix, so it is at (-4,7). The orientation of the parabola is vertical because the directrix is a vertical line. Therefore, its equation is of the form:
(x - (-4))^2 = 4p(y - 7)
where p is the distance between the vertex and focus (or vertex and directrix). Using the distance formula, we find:
p = |-3 - (-4)|/2 = 1/2
So the equation of the parabola is:
(x + 4)^2 = 2(y - 7)
Therefore, the standard form of the equation of the parabola is y = (1/2)(x+4)^2 + 7.