Question

I'm sure how to work the following problems cant anyone help?

1) Write an equation for the parabola with focus (4,0) and directrix y=2.

What do I need to do before I graph the following equations?

e1) 16x^2+9y^2=144

e2) x^2/9 - (y+2)^2/9 = 1

Answer 1

the focus is at y = 0 and the directrix is at y = 2 so the parabola opens down (sheds water.)
The distance from the focus to the vertex is the same as from the directrix to the vertex ( call it a ) so the vertex is at (4,1), halfway between directrix and focus
now so a = 1 but if parabola is upside down, use negative a so a = -1
with vertex at (h,k) which is (4,1)
form is
(x-h)^2 = 4 (a)(y-k)
(x-4)^2 = -4 (y-1)
x^2 - 8x + 16 = -4 y + 4
- 4 y = x^2 - 8 x + 16
y = -(x^2)/4 + 2 x - 4

Answer 2

e1) 16x^2+9y^2=144
this is an ellipse
(16/144)x^2 + (9/144) y^2 = 1
x^2/3^2 + y^2/4^2 = 1
center at (0,0)
x half axis 3
y half axis 4

e2) x^2/3^2 - (y+2)^2/3^2 = 1
hyperbola
center at (0, -2)
slopes of asymptotes = 3/3 = 1 (or -1)
vertex at (3,-2) and at (-3,-2)