The latus rectum measured on an open curve, by the ellipse's own calculation

0:00Here is a length worth having a name. Stand at one of a curve's two fixed points, draw the line straight across the main axis, and measure the piece of it the curve cuts off. That chord is what we are after. Two things about the definition before any algebra. First, it says either fixed point, so there are two such chords and not one. Over twelve curves built here, the number whose two fixed points give chords of different length is zero.

0:33Second, on an open curve those two chords sit on different pieces, one on each branch: sets where they landed on the same piece, zero. Take the closed curve first and pick one end of the chord. Its position across the page is the offset, because the chord stands over a fixed point. Call its height, half the chord, simply l. So the end of the chord sits at the offset across and l up.

1:04Now one rewriting, and it is the whole reason the answer looks the way it does. The offset is the ratio times the half-length along the main axis - that is what the ratio means. So write that end as the ratio times the half-length across, and l up. Nothing has happened yet. But the ratio is now in the working, and watching it leave again is the point of this whole topic.

1:34The end of the chord is a point of the curve, so it satisfies the curve's equation. Put it in. The first coordinate squared over the first half squared is the ratio times the half-length, all squared, over the half-length squared - and the half-lengths cancel, leaving the ratio squared. The second term is l squared over the second half squared, untouched. So the ratio squared plus l squared over the second half squared is one.

2:06Rearranged: l squared is the second half squared, times one less the ratio squared. That is a complete answer. It is also a bad one, and it is worth saying why. Look at what that answer needs. It needs the second half-length, which is sitting in the equation already. And it needs the ratio, which is not - to get the ratio you must first find the offset, and that is a whole extra step people reach for reflexively.

2:40Here that reflex is about to be shown up as unnecessary, because the ratio is not a new quantity at all. It is built from the two half-lengths, and if we say so out loud it will not survive to the end of the line. The ratio is the offset over the first half-length, so the ratio squared is the offset squared over the first half squared. And for a closed curve the offset squared is the first half squared less the second half squared.

3:13Put those together and the ratio squared is one, less the second half squared over the first half squared. Which means one less the ratio squared - the exact thing standing in our answer - is the second half squared over the first half squared, and nothing else. That is the identity the whole topic turns on, and it is one rearrangement of something you already had. Now substitute it back.

3:43l squared was the second half squared times one less the ratio squared. It is now the second half squared, times the second half squared over the first half squared. That is the second half to the fourth, over the first half squared. So l is the second half squared over the first half-length. And l was only half the chord. The curve is symmetric about the main axis, so the other half matches: over twelve curves, the number where the measured chord came out as the half rather than the whole is zero, and the number where it is exactly twice that half is twelve.

4:26The whole chord is twice the second half squared, over the first half-length. Look at what is not in that. No ratio, and no offset. Only the two numbers already written underneath the equation. A formula you have derived is still only a formula, so this one was put to work against curves built from the definition instead. Each curve here is a set of points: two fixed points and one constant, nothing else. The chord is then found by asking that set which points it holds on the upright line through a fixed point - and the answer comes out of the two distances alone, with no equation consulted.

5:12Twelve curves scored that way, six of each kind. Every one gives exactly two points on that line, and the number where the two heights are not each other's negatives is zero. Twenty-four chord ends, and the number where the distance the solve pinned down disagrees with the plane's own distance to that fixed point: zero. And the number of curves where the measured chord disagrees with twice the second denominator over the first: zero, out of twelve.

5:46The half-lengths are not read off anything either. The first is walked to along the axis; the second is measured on the line through the middle, which for the six closed curves gives the second denominator exactly - and for the six open ones gives nothing, because that line misses them. Now the open curve, and it is very tempting to stop here. The definition is the same shape, with transverse axis in place of major axis, and the answer turns out to be the same length. Why not just say so and move on?

6:26Because the derivation you have just watched used the closed curve's identity at one step, and the open curve's is not the same identity. Take the result on trust and you cannot rebuild it the day you misremember which of the ratio squared and one is the larger. Do the work once and there is nothing left to remember. The end of the chord still sits at the offset across and l up, and the offset is still the ratio times the first half-length.

7:01But the equation now subtracts, so substituting gives the ratio squared, minus l squared over the second half squared, equals one. So l squared is the second half squared, times the ratio squared less one. The gap is taken the other way round. That is the first reversal. Now the identity. For an open curve the offset squared is the two half squares added, so the ratio squared is one plus the second half squared over the first half squared.

7:37Which makes the ratio squared less one equal to the second half squared over the first half squared - the same expression the closed curve produced. That is the second reversal. Two sign changes, one in the equation and one in the identity, and they cancel each other. Over six closed curves the gap points below one every time and over six open ones it points above, but the size of that gap is the second denominator over the first for all twelve, with zero exceptions.

8:14So l squared is the second half to the fourth over the first half squared again, and the chord is twice the second half squared over the first half-length. Same answer, reached honestly - and you can now point at the step that carried the sign. It is worth seeing what happens if you take the gap the wrong way round. Use the closed curve's version on an open one, or the other way about, and the quantity you get is negative.

8:45Not slightly wrong - negative, and a squared length is not negative. Over all twelve curves, the number where the wrong way round leaves something that cannot be a squared length is twelve, and they all complain in the same words. So the arithmetic catches this by itself. You do not need to remember which way the gap goes; you need only notice when the answer stops being a length.

9:14Here is the sharpest way to see what the cancellation bought. Four open curves were built to share one ratio exactly - four different sizes of the same shape. Distinct values of the ratio squared among them: one. Distinct chords among them: four. So the ratio alone does not fix the chord. But take each chord against its own main axis, and the distinct values of that comparison among the four: one. The ratio fixes the shape of the answer, and the size has to come from somewhere else.

9:54Try the offset instead. Two curves built to share one offset: distinct offsets, one; distinct chords, two. The offset alone does not fix it either. Now group all twelve by their two denominators. Twelve groups, and the number of groups holding more than one chord is zero. That is what the cancellation says, said as a count. The chord needs both denominators, and it needs nothing else. One comparison before we finish, because it explains why this topic was harder than it looks.

10:33There is a third curve, built from one fixed point and one fixed line, and it has a chord of exactly this kind - reached without solving anything at all. Take the end of that chord. By the definition its distance to the fixed point equals its distance to the fixed line, and on the upright line through the fixed point that second distance is just the separation between the point and the line.

11:01So the half-chord is the separation and the chord is twice it. Over five such curves, the number where the endpoint's two distances differ is zero, and the number where the half-chord is not the separation is zero. No equation, and no identity. The two curves in this topic have no fixed line to lean on, so substitution is the only route open to them - and that is why their answer needs two letters where the third curve's needs one.

11:36One last comparison, and it retires a rule you may be carrying. For a closed curve the second half-length is always the smaller, so the chord is always shorter than the main axis. Over six closed curves, the number whose chord reaches past that axis is zero. For an open curve nothing forces that. Over six open curves, three have a chord longer than the axis - and they are exactly the three whose second denominator passes the first. The two counts pick out different curves zero times.

12:14One of them is worth drawing. Its axis through the two fixed points measures six; its chord measures eighteen. Three times as long. So a chord through a fixed point can be longer than the axis it crosses. Only on the open curve, and only when the second half-length wins. Using it takes almost no work, which is the point of a cancelled answer. Eighteen written equations were read here, every one needing to be divided through first. The number where the formula and the definition's own solve disagree about the chord is zero.

12:57The first gives eighteen fifths. An open one gives thirty-two thirds. One that hides its shape until you divide gives one half. And the long one from a moment ago gives eighteen. One of the eighteen has a first denominator of thirty-six fifths, which is not a square. Its chord cannot be written as a fraction at all - but its square can, exactly: eighty ninths. Of the eighteen, seventeen have a chord a fraction can name and one does not.

13:31Backwards is more interesting. Given the offset and the chord, the relation between the three lengths leaves a quadratic in the first half-length. Three such problems, six roots offered. Three are lengths; three are negative and get thrown away - not by a sign convention, but by the geometry, because a half-length is a length. The algebra is honest and offers both; you are the one who chooses. Build the curve back from the root that survives, measure its chord, and it returns exactly the chord you were given.