A parabola can be drawn given a focus of (11,8) and a directrix of x=5. What can be said about the parabola?
The parabola is vertical since the directrix x=5 is a vertical line.
The vertex of the parabola is halfway between the focus and the directrix, so it lies on the line x=8.
The distance between the focus and the directrix is the same for all points on the parabola. This distance is given by the equation |x-5|, where x is the x-coordinate of a point on the parabola.
Using the distance formula, we can find the equation of the parabola in vertex form:
d = |x-5| = sqrt((x-11)^2 + (y-8)^2)
Squaring both sides and simplifying:
(x-5)^2 = (x-11)^2 + (y-8)^2
x^2 - 10x + 25 = x^2 - 22x + 122 + y^2 - 16y + 64
Simplifying further:
12x = y^2 - 16y + 61
Therefore, the equation of the parabola in standard form is:
y^2 - 16y + 12x - 61 = 0