|
Question 581158: Please help me solve this problem: "Determine the equation of the parabola with the focus at (-4, 4) and where the directrix is the line y=-2. Find the two points that define the latus rectum and sketch the parabola."
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! Please help me solve this problem: "Determine the equation of the parabola with the focus at (-4, 4) and where the directrix is the line y=-2. Find the two points that define the latus rectum and sketch the parabola.
**
Equation is that of a parabola which opens upwards of the standard form: (x-h)^2=4p(y-k), (h,k) being the (x,y) coordinates of the vertex.(note that the directrix, y=-2, is below the y-coordinate, 4, of given focus)
Axis of symmetry:x=-4
y-coordinate of vertex=(4+(-2))/2=2/2=1 (midpoint formula)
x-coordinate=-4
vertex:(-4,1)
p=distance from vertex to directrix or focus on the axis of symmetry=3
4p=12
Equation: (x+4)^2=12(y-1)
..
latus rectum: a line passing thru the focal point perpendicular to the axis of symmetry connecting to both sides of the parabola at the following points:
y-coordinate for both points=4 (same as that of focus)
solving for x-coordinates:
(x+4)^2=12(y-1)
(x+4)^2=12(4-1)
(x+4)^2=36
x+4=±√36=±6
x=±6-4
x=-10
or
x=1
end points of latus rectum: (-10,4) and (2,4)
..
see graph below:
y=((x+4)^2+12)/12
|
|
|
| |
