So I was watching a video on Khan academy that explains how to find the equation of a parabola, given we have a focus and a directrix.

He draws a picture like this, the focus is at $(a, b)$ and we pick a random point on the parabola at $(x, y)$, the directrix is at $y = k$

The instructor indicates that we want to use the distance formula to help us with that.

enter image description here

What I don't understand, how can we apply the distance formula here? Where is the right triangle in that picture he drew?

  • $\begingroup$ The two line segment shall have equal length. And to compute length you need the said formula $\endgroup$ – user122049 Dec 22 '17 at 9:06
  • $\begingroup$ @user122049 ok, so the pink line segment and the thicker, blue line segment have the same length, ok I understand that. But I don't get on which line segments he applies the formula. $\endgroup$ – Max Dec 22 '17 at 9:09
  • $\begingroup$ He applies them to those segments! Blue - distance between the points with coordinates $(x,y)$ and $(x,k)$. Pink - distance between the points with coordinates $(x,y)$ and $(a,b)$. This is the definition of a parabola: the set of all points with equal distances between a point and a line. $\endgroup$ – Nicky Hekster Dec 22 '17 at 9:14

You don't need to draw a right triangle to use the distance formula. But if you did draw one, the one being measured has the line segment $(x,y)$ to $(a,b)$ as its hypotenuse. The right angle would be at $(a,y)$ (Draw a vertical line through $(a,b)$, a horizontal line through $(x,y)$, and mark the intersection).

The expression on the left is the distance from the point $(x,y)$ to $(x,k)$. There's no right triangle to draw there, because one of its legs would have zero length.

  • $\begingroup$ Oh, I got it now, thanks for the clarification. The second paragraph is still a bit unclear, if one leg has 0 length, it's still a triangle? $\endgroup$ – Max Dec 22 '17 at 9:25
  • $\begingroup$ @Max When one side length is 0, it’s not a triangle at all! $\endgroup$ – Rohan Dec 22 '17 at 9:27
  • $\begingroup$ Ah I think I need to think a bit more about groking the whole thing here, but you already cleared up a lot of questions I had. Thank you! $\endgroup$ – Max Dec 22 '17 at 9:50

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