Once c outgrows a, the ellipse's eccentricity and equation become the hyperbola's

0:00Almost nothing here is new. You already have a curve built from two fixed points and one constant, and the derivation that turns that sentence into an equation. This time the constant is a difference rather than a total, and one comparison between two lengths comes out the other way round. Everything that follows comes from that reversal. The algebra clears the same two square roots, but one quantity in the answer has gone negative, so it gets a new name and the plus becomes a minus.

0:36Even the closing move is the same manipulation run twice: one version boxes the curve in, the other empties a band out. So do not start again. Run the argument you have and watch where it forks. Start with the ratio. It is the offset divided by the half-length along the axis carrying the two fixed points - the same two numbers, in the same order. For the closed curve the offset was the smaller, so the ratio came out below one every time. Here the offset is the larger, and the ratio cannot come out below one at all.

1:19Six frames of each kind were built and scored. Six of the six open frames have a ratio bigger than one, none equal to one and none below it; six of the six closed frames come out below one, and none reaches one. Four distinct ratios on each side: thirteen twelfths, five quarters, five thirds and thirteen fifths for the open curve, and their reciprocals for the closed one. There is a looseness here worth naming rather than repeating.

1:55It is easy to say the offset is at least the half-length, as though the two were allowed to be equal. They are not. If they are equal, the second half-length is the square root of nothing, and the equation you are about to derive divides by it. The algebra says so on its own: run the derivation with the offset set equal to the half-length and what it leaves on the right-hand side is nothing at all, so there is nothing to divide through by.

2:28The condition the argument actually needs is the strict one: the offset must be bigger than the half-length, not merely as big. That single word is the difference between a curve and a division by zero. Now the frame, and there is nothing to decide. The two choices are the ones you made for the closed curve, made again: the middle of the two fixed points at the origin, and the line through them along the horizontal axis.

3:01The fixed points sit at minus the offset and plus the offset, and the definition reads: the farther distance less the nearer one is the constant, which is twice the half-length. Nothing in that setup knows which of the two numbers is larger - which it must not, if this is to be one argument. Here is the step that makes the topic. The definition has two square roots in it. Move one across, square both sides, and a single root is left by itself. Square again, and none are left.

3:39Written on the squared distances, that whole chain is one line: four times the constant squared, times the second squared distance, equals the first squared distance less the second less the constant squared, all squared. Look at what is missing from that line. The sign. Whether you moved the root across a plus or a minus, the first squaring destroys the difference between them - so the same line serves both conditions, run once and read twice.

4:14The term in the first coordinate on its own cancels, appearing on both sides with the same coefficient, and what survives is a term in each square and a number. Over every frame tested, the count of leftover terms that should not be there is zero. So where does the difference between the two curves actually enter? In one number. Divide through and the first denominator is the half-length squared, and the second is the half-length squared less the offset squared.

4:50For the closed curve the half-length is the bigger of the two, so that is positive. For this one the offset is bigger, so it is negative. Run the algebra on the frame with half-length four and offset five, and the two denominators come out sixteen and minus nine. Run it on the closed frame with half-length five and offset four, and they come out twenty-five and nine. Add those two second denominators and you get nothing. They are the same quantity with opposite signs, and that is the entire difference between the two curves.

5:30A negative denominator is awkward to write, so name the positive quantity instead: nine is three squared, so call it three. Now the second term carries a minus in front of it - and nothing has changed. You have only stopped writing a negative number as one. Put the two results side by side. The closed curve: the first coordinate squared over the first half squared, PLUS the second coordinate squared over the second half squared, equals one.

6:05The open curve: the first coordinate squared over the first half squared, MINUS the second coordinate squared over the second half squared, equals one. One symbol. That is the difference between a closed loop and a pair of branches running away from each other forever, and it is the only difference there is. None of that is worth much unless the equation and the definition describe the same places, so they were scored against each other - the equation kept as a separate object, knowing no fixed point and no constant, so it is free to be wrong about the set.

6:48Twelve curves, six of each kind, and two hundred and sixty-four exact places on them, each put to the definition and to the derived equation separately. Places where the two disagreed: zero. Places the definition itself rejected: zero. The same places nudged a seventh of a unit off the curve and offered to both: accepted by either, zero. Over a window of two thousand nine hundred and ninety-three places, six hold the open curve and the same six hold its equation; four hold the closed curve and the same four its equation.

7:31And to show the sign is doing real work, take the same equation with the sign left unchanged. It holds at four window places, two of them on the open curve and two not, and the two disagree about six places in all. Now go the other way: take a place that satisfies the equation and recover its two distances. Substitute for the second coordinate and expand, and both squared distances turn into perfect squares.

8:05The first is the half-length plus the ratio times the first coordinate, all squared. The second is the half-length minus the same thing, squared. Both distances are linear in the first coordinate - the same surprise as before, and the same reason the difference cannot move. Twenty-one exact places were walked. Places where the first squared distance is not the first bracket squared: zero, and the same for the second one.

8:39Take a place on the branch nearer the second fixed point: five across, nine quarters up. Its two distances are forty-one quarters and nine quarters, differing by exactly eight - twice the half-length. And notice they are nowhere near equal. The constant is not two halves of anything. But a squared bracket does not give you the distance. It gives it up to a sign, and here the sign changes.

9:10Of the twenty-one places, eleven sit on the branch nearer the second fixed point and ten on the other, and none on neither. On the near branch the first bracket is positive and the second negative, at all eleven; on the far branch those signs are exchanged, at all ten; and at none of the twenty-one is a bracket exactly nothing. So on one branch you take the first distance less the second, and on the other the second less the first - and both give twice the half-length. Places where the subtraction the other way round also works: zero.

9:51That is not a new rule. It is the convention from the last part, showing up in the algebra as a sign. Which fixed point is the farther, counted directly: the first at eleven places, the second at ten, neither at none of them - and over the whole window, forty-one places where neither is, all of them on the line down the middle. One more thing to watch: it is easy to write that ratio upside down here, and the answer still looks like a distance. Places where the second squared distance matches that version: zero.

10:31Now the closing manipulation, the same one both times: rearrange to put the first term alone. For the closed curve you get a quantity that is at most one, because you are subtracting something that cannot be negative. For this one you get at least one, because you are adding it instead. Same move, opposite conclusion, and the only input that changed was the sign of the second term. The closed reading gives a box: neither coordinate leaves its range, and the curve lives inside a rectangle.

11:09The open reading gives the opposite: a band the curve is banned from. The size of the first coordinate is at least the half-length, so everything strictly between the two vertices is empty. Count both. Twenty-three exact places on the closed curve, and places among them outside its own box: zero. Of the window's two thousand nine hundred and ninety-three places, one thousand and twenty-five are inside that box and one thousand nine hundred and sixty-eight are outside it, and of the ones outside, the number the closed curve holds is zero.

11:53Now the band. One thousand two hundred and seventy-one window places lie strictly between the two vertices, and the number of those the open curve holds is zero. The band is eight units wide and empty. One detail is easy to lose, and losing it costs you half the curve: the conclusion is about the SIZE of the first coordinate, not the coordinate itself. Read it one-sided, without the bars, and it keeps eleven of the twenty-one exact places and throws away ten - and every one of those ten is a genuine place of the curve.

12:36The two fixed points do not have to lie across the page. Put them up and down instead and every step runs the same, with the two coordinates exchanged. The equation becomes the second coordinate squared over the first half squared, minus the first coordinate squared over the second half squared, equals one - and the branches open upwards and downwards. Scored the same way: six window places, and zero disagreements between that equation and its definition.

13:12So how do you tell which way up a given equation is? By the sign, and only by the sign: whichever squared term carries the plus, its denominator names the axis carrying the two fixed points. Over the six open frames the sign reads the axis correctly six times out of six, and over the six closed frames it correctly names no axis at all - both terms are positive, so neither is picked out.

13:45And now the mistake this part exists to prevent. For the closed curve there was a shortcut: the larger denominator belonged to the longer axis, every time. That is a theorem there - the half-length through the two fixed points really is the larger of the two. It is not a theorem here, and not even usually true. Over the six closed frames the larger denominator names the wrong axis zero times. Over the six open frames it names the wrong axis three times out of six.

14:23The reason is that nothing here forces the second half to be smaller than the first. Closed frames whose second half beats the first: zero. Open frames whose second half beats the first: three. In all six open frames the offset is the largest of the three, and in all six closed frames the first half is. Frames of either kind obeying the other kind's relation: zero. The biggest of the three sits alone on its own side of the relation, and which one that is has swapped.

15:02One case is worth ending on, because it is the only one where the ratio is forced. Make the two halves equal, and the offset squared is twice the half-length squared, so the ratio squared is exactly two - for every such curve, at every size. Four such frames were built at four different sizes: all four give a ratio squared of two, and the number of distinct values among them is one.

15:32The ratio itself is not a fraction. Of the four, the number whose ratio is a rational length is zero and the number whose offset is one is zero as well - which is why none of the six open frames built from whole numbers has equal halves. So what has changed since the closed curve? One comparison between two lengths. From it: a ratio that cannot fall below one, a quantity that goes negative and gets renamed, a plus that becomes a minus, a box that becomes an empty band, and a reading rule that stops working - while the derivation did not change at all.