Paraboloid Mathematics Questions
This page is written in response to Mr. Bridger's page on Paraboloid Mathematics.
1) The vertex must lie on the Y-axis because points on one side of the vertex will be the same distance as points with the same Y value on the other side; this means that the parabola will be symmetrical across the vertical line going through the focus. Parabolas are symmetrical across the Y-axis, so the Y-axis must go through the focus.
The directrix line must be y = -a for two reasons. One, the directrix must be perpendicular to the axis of symmetry, for reasons very similar to those explained above. The axis of symmetry is, in this case, the Y-axis; the equation y = -a, when graphed, creates a line paralell to, and a units down from, the X-axis. A second reason is that the distance from the directrix from the vertex must be the same as the distance from the vertex to the focus. The closest point along the directrix to the vertex ( (0, 0) ) is (0, -A); the distance there is A units down. The focus is A units up from the vertex, so the distance is the same.
The distance between any given point (x, y) on the parabola must equal y + a because all points (x, y), by definition, are y points above the X-axis, and the directrix is A points below the X-axis. Therefore, to get the total distance, just add y and A, and you get y + a.
Picture altered from the lab page
Given that the distance from a point on a parabola to the focus and the same point on the parabola to the directrix must, by definition, be the same, the equation x2 = 4ay can be proven fairly simply:
Length = y + a = Ö(Dx2 + Dy2)
y + a = Ö(x2 + (y - a)2)
(y + a)2 = x2 + (y - a)2
y2 + 2ay + a2 = x2 + y2 - 2ay + a2
y2 + 2ay + a2 - (y2 - 2ay + a2) = x2
y2 + 2ay + a2 - y2 + 2ay - a2 = x2
2ay + 2ay = x2
4ay = x2
x2 = 4ay
2) I will use the following technique to determine A, the focal length of the dish: I will measure the diameter of the dish, and the height from the vertex of the dish to the rim. The radius, or half of the diameter, and the height represent one (x, y) coordinate set that is on the parabola. I can therefore take the the diameter, halve it, and plug it in for X, and take the height and plug it in for Y, and solve for A in the equation derived above (X2 = 4AY). Solving for A, (X2)/4Y = A.
Picture altered from the lab page
I measured the height to be 14cm, and the diameter to be 30cm. Therefore, applying the formula derived above,
X2 / 4Y = A
(30)X2) / 4(14) = A
900 / 56 = A
16.1cm = A
The focal length of this parabolic dish is 16.1cm., or roughly 2cm above its top edge.