A New Math Equation I Discovered

It’s actually three equations. They are depicted above. What do they do? They are for an up-down parabola. If we take a line of slope m through the focus of a parabola with parabolic constant a, then the distance d between the intersection points is split up by the focus of the parabola into two parts d1 and d2.

So, the top equation is the easiest one to understand — the distance d between the intersection points is simply the sum of d1 and d2. The second equation lets you calculate the parabolic constant a in terms of d1 and d2. If you want an upward parabola, choose a to be positive — downward means a is negative.

Here’s something special — note that there is something very different about the third equation from the first two equations — the presence of the square root.

Yes, you can calculate the slope of the line with a given d1 and d2, but because of that pesky square root, you won’t always get a rational m even when both d1 and d2 are chosen as rational numbers. What needs to happen?

The product of d1and d2 needs to be a perfect square to get a rational m. This is a huge idea — so much so that I called it a theorem in my math book.

Ok. We will make up an example. Let us choose a couple of lengths for d1 and d2. Let d1 = 2*(5/3) and d2 = 2*(3/5). Note that 5/3 and 3/5 are reciprocals, so their product is one, which is a perfect square. These fractions are each multiplied by 2, and 2x2 = 4, which is a perfect square. The product of perfect squares…