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ducmod
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Homework Statement
Hello!
Please, help me to get through equations. I can't derive the equation in the way suggested.
Here is the definition:
If we choose to place the vertex at an arbitrary point (h; k), we arrive at the following formula
re-deriving the formula from Denition 7.3. (If the vertex is at (0;0), then from the definition of parabola,
we know the distance from focus point (0; p) to a point (x; y) is the same as the distance from a point
on directrix (x;-p) to the same point (x; y)).
Using the Distance Formula we get:
The Standard Equation of a Vertical Parabola: The equation of a (vertical) parabola with vertex (h; k) and
focal length |p| is
(x - h)^2 = 4p(y - k)
If p > 0, the parabola opens upwards; if p < 0, it opens downwards.
So, if vertex is at (h, k) and there is a given point on a parabola at (x; y), then focus is at (h; k + p),
and point of a directrix is at (x; k - p).
Thus the distance formula should be (I am dropping the square root):
(x - h)^2 + (y - k)^2 = (y - k + p)^2
How did they come to (x - h)^2 = 4p(y - k) ?
Thank you!