Balancing a point against a line, and the four equations that result

0:00A circle needs one point and one number. A parabola needs a point and a line. That looks like a small change, and it is not. Two points, or a point and a number, can always be pushed into the middle of the picture and the answer will not care. A point and a line cannot. They sit relative to each other, and how they sit is now part of the problem.

0:26So this topic has a step the circle never had: before anything is written down, the axes have to be chosen. And the word standard, in standard equation, turns out to name that choice rather than anything the curve itself possesses. The curve does not know where your axes are. Here is the definition, and it is one sentence. Every place of the plane that is exactly as far from one chosen point as it is from one chosen line, where the point is not on the line.

1:02Draw it and you get two segments from every place of the curve: one to the point, one square across to the line. The curve is where those two lengths are the same. That is worth building rather than believing, so it was built: the point, the line, and every place tested by comparing the two lengths. Across the places of the curve, the number where the two drawn segments came out unequal is none.

1:31And every foot the second segment lands on really is a place of the line, which is what makes it the distance and not just a distance. No equation yet. There is nothing here but places and lengths. The curve does more than sit there. It cuts the plane into two sides. Over a window of three thousand two hundred and one places, one thousand three hundred and sixty-eight came out nearer the point, one thousand eight hundred and sixteen nearer the line, and seventeen exactly the same distance from both.

2:09Those seventeen are the curve, and the three counts add back to three thousand two hundred and one, so nothing was double-counted and nothing was lost. Now the names, and each one is a thing you can point at. The chosen point is the focus. The chosen line is the directrix. The line through the focus square to the directrix is the axis, and the place where the curve meets its own axis is the vertex.

2:40Two things are worth saying out loud because they are commonly got wrong. The focus is not a place of the curve, and the directrix never touches it: the number of places on both the curve and the directrix is none. The definition carries an exclusion. The point must not lie on the line. It is easy to read that as bookkeeping. It is not. The question still makes sense if you break the rule, so it was asked: put the point on the line and ask which places are the same distance from both.

3:18The answer came back as ninety-seven places, and they are exactly the line square to the directrix through that point. Not a curve. A straight line. Places the test found that are not on that straight line: none. Places of it the test missed: none. So the exclusion is not tidiness. It is the boundary between a parabola and something else entirely. The same count came back in a picture that had been turned and slid, so it is a fact about the two ingredients rather than about the page.

3:56Now the choice of axes, made in the open. Drop the perpendicular from the focus to the directrix. That segment is the only thing in the picture the two ingredients agree on. Take its midpoint. Put the origin there. Run the first axis along that perpendicular, pointing towards the focus, and the second axis square to it. Every one of those decisions buys something. The midpoint is on the curve, because it is the same distance from the point and the line, so it is the vertex and not some arbitrary place.

4:35Putting the axis along the perpendicular makes the focus sit at a along the first axis and nothing along the second, and puts the directrix at minus a. Nothing has been derived yet. All that has happened is that a frame was built, out of the two objects that were given. Now let the definition speak. Take a place, x and y. Its distance to the focus is the distance formula. Its distance to the directrix is how far it sits horizontally from that vertical line.

5:14Set the two equal, square both sides, and expand. On the left, x minus a, squared, plus y squared. On the right, x plus a, squared. The x squared cancels. The a squared cancels. What is left is minus two a x on one side and plus two a x on the other. y squared equals four a x. That is the whole derivation. The squaring is the one step that needs watching, and here it costs nothing, because both sides were distances before they were squared, and a distance is never negative.

5:53One line. No trick. The definition, written in coordinates. Here is the claim this whole topic rests on, and it is easy to say and easy to doubt. The four standard equations are not four curves. They are one argument seen from four sets of axes. So it was measured. Sixty curves were built - four placements, five lengths, three standings of the picture - and each one was given the equation its own focus and directrix produce.

6:27In the frame each picture happened to be drawn in, only twenty of the sixty came out with just two terms. The other forty carried more. Then the frame was built from the focus and the directrix, by the recipe, and the same sixty curves were written out again. This time all sixty came down to two terms. None carried more. And those sixty equations are only five different equations, one for each length.

6:58Four placements, and in their own frames they are the same equation. Standard is a fact about the axes, not about the curve. There is a second half to the argument that is easy to skip, and skipping it is skipping the proof. The derivation shows that every place of the curve satisfies the equation. It does not show the other direction. Left there, the curve is sitting somewhere inside the equation's solutions, and the equation might be describing more than you meant.

7:33So substitute back: put four a x in place of y squared in the distance to the focus and you get x plus a, squared, which is the distance to the directrix. Every solution is a place of the curve. To show that is a real risk and not a formality, a second equation was carried alongside: the true one multiplied by its own mirror image. Every place of the curve satisfies it. Zero failures forward, for both.

8:07But going the other way, the true equation has no solutions off the curve, and the second one has ten. Forward agreement alone would have called them both correct. One letter in this topic does three jobs, and students conflate them constantly. a is a length. It is the distance from the vertex to the focus. It is also the distance from the vertex to the directrix, because the vertex is halfway.

8:39The distance from the directrix to the focus is therefore two a. And the number standing in front of x, in the equation, is four a. With a equal to three: the vertex is three from the focus, the focus is six from the line, and the number in the equation is twelve. Three different numbers. One letter. Across every length tested, the number of times those four quantities did not come out as a, a, two a and four a is none - so the relationship is measured, not remembered.

9:16The equation says something the picture does not obviously say. y squared equals four a x, and a is a length, so a is positive. y squared is never negative. So four a x is never negative. So x is never negative. Of the places of this curve, the number with a negative first coordinate is none. Half the plane is simply out of reach. Exactly one place sits on the upright axis, and that is the vertex.

9:48The mirror follows from the same equation. y appears squared, so plus y and minus y give the same x. Of the seventeen places of the curve, all seventeen reflect across the flat axis back onto the curve. Across the upright axis, exactly one does, and that one is the vertex again. A squared y means the flat axis is the mirror. It is worth reading the argument rather than the letter, because the letter suggests the opposite.

10:25A word about how any of this was checked, because there is a way of checking that proves nothing. If a curve is defined by its equation, then testing the equation against the curve is testing a thing against itself. So the curve here is never an equation. It is a set of places, built by comparing two distances, and the equation is a separate object scored against it. Sixty curves, each equation asked about every place of a window: a hundred and ninety-two thousand and sixty scorings.

11:03Places where the set said one thing and the equation said the other: none. And to show that scoring can catch a wrong equation, each curve was also given a decoy with its directrix nudged one step. All sixty decoys are close enough to the window to be visible, and all sixty were caught. None slipped through unseen. That is the difference between an equation that agrees with itself and an equation that is right.

11:35Given an equation, you should be able to say what the curve looks like without drawing anything. Two questions. Which variable is squared, and what sign does the other one carry. A squared y means the flat axis is the mirror. A squared x means the upright axis is. With the flat axis as the mirror, the curve opens right when x carries a positive number and left when it carries a negative one; with the upright axis, up or down by the same rule.

12:10That reading was written as a routine that sees the coefficients and nothing else - no focus, no directrix, no places. Twenty equations were read that way and each answer was checked against the set, which was measured separately. Disagreements: none. Four distinct answers came out of twenty equations, which is the four placements arriving again from a different direction. And handed an equation from a turned picture, the reading refuses rather than guesses, because that equation is not one of the four shapes.

12:50Now the questions run the other way. You are given a condition and asked for the curve. Focus two along the flat axis, directrix the vertical line two back. Both ingredients handed over, so the derivation just runs: y squared equals eight x. Vertex at the origin, focus two up, and no directrix mentioned. But the directrix is not free - it has to be as far from the vertex on the other side, square to the line joining them. That puts it at minus two, and the equation is x squared equals eight y.

13:31Now a harder one. A curve mirrored by the upright axis, passing through the place two along and three down. The place is below the axis, so the curve must open downwards. Fixing the sign before any arithmetic is the whole content of the question. Searched over every length on a fine grid and both upright placements, the number that fit is exactly one: opening down, with a equal to a third.

14:02Two more of the same kind give a equal to nine eighths and to twenty-five eighths. Neither is a whole number, which is the reminder that a is a length, not a tidy coefficient. Three questions from outside mathematics, and one of them is a trap. A beam twelve across between its supports sags three centimetres in the middle, in this shape. The number in front of the coordinate comes out as twelve hundred.

14:34Where has it sagged one centimetre less than the deepest? One centimetre less than three is two centimetres above the lowest place, not one, and the squared distance out from the middle is twenty-four. Substitute the one centimetre straight in and you get twelve instead. The figure is what tells you which number to use. An arch ten tall and five across at its foot gives five eighths, and two down from the top its full width squared is five.

15:07A cable a hundred across, thirty at its ends and six in the middle, gives six hundred and twenty-five sixths, and the wire eighteen out is five thousand six hundred and ninety-four over six hundred and twenty-five. And a last honesty. All four standard forms need the vertex at the origin and the focus on an axis. A curve with a focus and a directrix anywhere else is a real parabola, and none of these four equations can name it.