A parabola can be drawn given a focus of (3,−5) and a directrix of y=−11. What can be said about the parabola?
The parabola is downward-facing and its vertex is at the midpoint between the focus and the directrix. Using the formula for the distance from a point to a line, the distance from the point (3,−5) to the line y=−11 is:
d = |−5−(−11)|/sqrt(1) = 6
This distance is also the distance from the vertex to the focus (or directrix), so the vertex must be at:
(3,−5−6/2) = (3,−8)
Therefore, the equation of the parabola is:
(y − (−8)) = −1/4(x − 3)^2
And we can see that the parabola is downward-facing (since the coefficient of x^2 is negative) and opens towards the focus (since the focus is below the vertex).