The chord through the focus that measures how open the curve is

0:00Every parabola carries a number called a, and you cannot point at it on the curve. It runs from the deepest place of the curve out to the focus, and the focus does not sit on the curve at all. So a is a distance between a place you can see and one you cannot. It is worse than that. Four different lengths in this picture all wear the letter a.

0:25Deepest place to focus is a. Fixed line to focus is two a. Focus to one end of a certain chord is two a again. And that whole chord is four a. Take a equal to three, and those four come out three, six, six and twelve. Four lengths, three different numbers, one shared letter. That is why they get swapped for one another so freely. This is about the last of the four, the chord of length four a, the only one you can lay a ruler along.

1:00The chord has a name. It is called the latus rectum. And its definition asks for three things at once, not one. It has to pass through the focus. It has to be square to the axis of the curve. And both of its ends have to lie on the curve. Drop any one of those and you are talking about a different chord. Here is what each condition is worth, counted.

1:26Take five curves, each stood three ways, and pair up ten named places on every one. That gives six hundred and seventy five pairs. All of them have both ends on the curve, because that is how they were built. Seventy five of them run through the focus. Seventy five of them are square to the axis. But only fifteen do both. A hundred and twenty pairs satisfy exactly two of the three conditions, and every one of those is an impostor.

2:01Before the argument can start, one length has to be nailed down. How far is the focus from the fixed line? When the frame was chosen, the deepest place of the curve was put at the origin, exactly halfway between the two. The focus went to a along the axis. The fixed line went the same distance the other way. So the gap from the line to the focus is a plus a. Two a.

2:29That number is not measured off a drawing. It is a consequence of where the origin was put. And it is the only thing the argument will borrow from outside the one sentence that says what a parabola is. Now the figure the whole argument lives in. F is the focus, sitting on the axis. A is the upper end of the chord, and B the lower end. M is the foot of the perpendicular dropped from the focus onto the fixed line.

3:02And C is the foot of the perpendicular dropped from A onto that same line. Five letters, and one of them does all the work. Look at what A F, A C, F M and the fixed line make between them. Two sides run along the axis direction, two run along the line, and the corners are square. The shape A C M F is a rectangle. That is rarely said out loud, and the whole argument turns on it, so we will say it.

3:35First move. A is a place on the curve. The definition of a parabola says exactly one thing about a place on the curve. It is as far from the focus as it is from the fixed line. A C is that second distance, measured straight across. So A F equals A C, and no equation was touched. Second move. A C M F is a rectangle, so A C equals F M, the side opposite it.

4:05Third move. F M is the gap from the fixed line to the focus, which we pinned at two a. Chain them. A F equals A C equals F M equals two a, and that was forced by the definition before any coordinates existed. Sixty curves were put through those three moves, one at a time. The number where an end was not on the curve, the number where the definition step failed, and the number where the rectangle was not a rectangle, are all zero.

4:40Half the chord is done. The other half costs nothing. A parabola is a mirror image of itself across its own axis. The chord is square to that axis, so reflecting the picture sends A to B and leaves F where it is. Whatever A F is, F B is the same. Two a on one side, two a on the other, so the whole chord is four a. That is the result, and nothing in it came from an equation.

5:09One more thing the picture is quietly telling you. The chord runs parallel to the fixed line, never across it. Over all sixty curves, the number of chords not parallel to the line is zero, and the number that reach it is also zero. It sits two a away from the line and stays there. There is a second way to the same number, and it takes one line. The chord sits at the focus, so its places have first coordinate a.

5:42Put that into the standard equation. y squared equals four a times a, which is four a squared. So y is plus or minus two a, the half chord is two a, and the whole chord is four a. Same answer. That was checked as a search rather than as algebra. For five curves, every height where the equation comes to nothing at the focus was hunted down. The number that found the wrong count of places is zero, and the number that disagreed with the length measured on the set is zero.

6:18And to be sure the search can tell chords apart, one curve was cut well past the focus instead. It found a chord, and it was not this one. Two routes, one answer. Be clear about what each one is for. The substitution is quicker, and it confirms the number. But it explains nothing. It tells you that y comes out two a, not why. And it can only run once an equation exists, which means once someone has already chosen where to put the axes.

6:53The argument from the definition never touches an equation. It says the end of the chord is as far from the focus as from the line, and that distance was fixed at two a when the frame was built. So the length is settled before any coordinate system is written down. Move the whole picture, turn it, tilt it, and four a survives, because none of the three moves mentioned a coordinate.

7:22One route explains. The other checks. Keep both, and know which is which. Now the practical payoff. In the standard equation, the number multiplying the plain variable is four a. And four a is the chord. So the length is not something you compute. It is the size of a coefficient already sitting in front of you. That claim was scored, not assumed. Sixty equations were handed to a routine that sees six numbers and nothing else, no focus and no curve.

7:58The number of times its answer disagreed with the chord measured on the actual set of places is zero. And the number whose answer could not be built back into its own equation is also zero. Write the same equation three times as large and the reading does not move. Eight either way. It has limits, though. Set the mirror of the curve at a slant and the equation grows a mixed term, and a constant that will not cancel.

8:30Four equations of the wrong shape were offered to it. The number it read a length out of anyway is zero, and it gave two distinct reasons for refusing. It refuses rather than guesses. One reading, done slowly. y squared equals eight x. y is the squared variable, so the curve is a mirror image of itself across the flat axis. x carries a positive coefficient, so the curve opens to the right.

9:03Match it against the standard shape. Four a equals eight, so a equals two. The focus is a along the axis, at two, nothing. The fixed line is the same distance the other way, at x equals minus two. And the chord is four a, which is eight, the coefficient we started from. Notice the last step took no arithmetic. It was already written down. Six equations, one procedure, and a pattern the set is built to expose.

9:37Their values of a come out three, three halves, two, four, five halves, and nine quarters. Their chords come out twelve, six, eight, sixteen, ten, and nine. Three of the six values of a are fractions. None of the six chords is. That is the whole reason a is awkward and the chord is not. The chord is the coefficient; a is that coefficient split four ways. A fractional a is not a mistake. It just means the coefficient was not a multiple of four.

10:12And in none of the six does a equal the chord, which is the swap that costs the most marks. Remember the chord rather than a. It is the one on the page. Two places where this stops being bookkeeping. A curved mirror, its focus five out from the deepest point, hollowed to a depth of forty five. How wide is its open end? The width was not assumed. The two places on the rim were searched for, and the span across them comes out sixty.

10:45Now the same dish backwards. The rim is twenty across and five deep, and you are asked where the focus goes. Searching every candidate length, exactly one puts the rim on the curve, and it is five. Same dish, opposite direction. And one more. Take the deepest place of a curve and the two ends of its chord, and you have a triangle. For the curve whose coefficient is twelve, a is three, the ends sit at minus six, three and six, three, and the area enclosed is eighteen.

11:22So, to carry away. The chord is four a, and that was settled by the definition alone, before any equation existed. But once an equation exists, that number is already sitting in it as a coefficient. The length you could not see is the number you were looking at all along.