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2. Where is the touching point?

Where does the string touch the parabola? See if you can come up with your hypothesis!

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Video transcript

- We're ready for step two in our search for the elusive formula of how to compute a point exactly on a parabola. To gain some intuition, let's go back to the interactive that we used to kick this all off. So here's my string art construction with a bunch of lines. And now, let me cycle through the lines one at a time, highlighted here in magenta. So there's one of them, here's another one, here's another one. And each of these magenta lines can be described by some fraction T that names that point on the line. As I wiggle the magenta line back and forth, notice that it continues to touch the parabola in exactly one place. Well if I can develop a formula for where that place is, that's the formula I'm looking for. We can see that even more clearly in this version of the interactive that you'll be using in a moment to solve a couple of exercises. So in this version, I can use this perimeter T to name which string art line I'm talking about. And notice that each of the string art lines touches the parabola in exactly one place. In the next exercise, you'll be asked to come up with a hypothesis for where that place is. And the hypothesis, as a hint, has something to do with the ratios of these subsegments that are being created. And we've color coded those subsegments to give you an even more of a hint. So see if you can figure out what that special relationship is.