Find an equation for the parabola with focus at (-8,-4) and vertex at (6,-4)
i do not know how to figure this one out. will you please help me.
i dont know what to do. thank you a lot.
aloha!
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The equation of a parabola with vertex (h,k) is either
(x - h)² = 4p(y - k)
where the vertex is (h, k),
|p| = distance from vertex to focus,
p positive if focus is above vertex (opens upward), and
p is negative if focus is below vertex (parabola opens downward).
focus (h, k+p), and
directrix is the horizontal line whose equation is y = k-p
or
(y - k)² = 4p(x - h)
where the vertex is (h, k),
|p| = distance from vertex to focus,
p is positive if focus is right of vertex, (parabola opens to the right)
and p is negative if focus is left of vertex, (parabola opens to the left)
focus (h+p, k), and directrix is the vertical line whose equation is
x = h-p
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Since the vertex is (6, -4), h = 6 and k = -4
Since the vertex (6, -4) and focus (-8, -4) have the same y-coordinate,
it is of the second type.
The distance from vertex to focus is 14 units. Since the focus is left
of the vertex, p is negative, and so p = -14. (Parabola opens to the left)
So the equation is
(y - k)² = 4p(x - h) or
(y - (-4))² = 4(-14)(x - 6)
(y + 4)² = -56(x - 6)
If you were asked for the equation of the directrix,
it would be x = k-p or x = -4-(-14) = -4+14 or x = 10
Edwin