Question

I need help, medals given :) Pleasee

Answer 1

Derive the equation of the parabola with a focus at (2, 4) and a directrix of y = 8.
A. f(x) = −one eighth (x − 2)2 + 6
B. f(x) = one eighth (x − 2)2 + 6
C. f(x) = −one eighth (x + 2)2 + 8
D. f(x) = one eighth (x + 2)2 + 8

Answer 2

well the general equation I use is

\[(x - h)^2 = 4a(y - k)\]

the directrix is above the focus so the parabola is concave down

so a is negative (a is the focal length, distance between the focus and vertex)

the distance from the focus to the directrix is 2a

so looking at the y values

2a = 4 - 8

so a = - 2

if the focus is (x, y) then the vertex is (x, y - a)

so in this case its (2, 4 - -2) = (2, 6)

so the vertex is (2 6)

now substitute into the general form

\[(y - 2)^2 = 4 \times -2(y - 6)\]

making y the subject

\[y = -\frac{1}{8}(x - 2)^2 + 6\]

you can also find the equation by selecting a point on the curve P(x, y)

and the using the fact that the point is equi distance from the focus and directrix.

and use the distance formula to show it.


hope it helps

Answer 3

Wow, thank you sooo much! That massively helps :) @campbell_st