Putting the centre at the origin, and reading the axes off the equation

0:00An ellipse does not come with an equation attached. A curve is a set of places. An equation is a sentence about numbers. To get from one to the other, you have to say where you are standing. So before any algebra, two decisions get made. First, put the middle of the curve - the halfway place between the two fixed points - at the origin. Second, lay the line through those two fixed points along one of the coordinate axes.

0:31Neither decision changes the curve. You can slide a picture and turn it as much as you like, and the shape is untouched. What the decisions change is how much work the algebra has to do. Put the middle anywhere else and every expression carries two extra letters. Turn the axis away from the coordinate axes and a cross term appears. What they buy is an equation short enough that you can read the geometry straight off it.

1:04With those two decisions made, the picture is pinned down. The two fixed points sit at minus c, zero and at plus c, zero. Call the constant that the two distances add to two a. Writing it as two a is deliberate: a will turn out to be half the long axis, and there will be a reason. Take any place P, with coordinates x and y, on the curve.

1:31Its distance to the left point is the root of x plus c, squared, plus y squared. Its distance to the right point is the same thing with a minus. The definition says those two roots add to two a. Everything from here is arithmetic on one sentence. Two square roots, one equation, and a promise that the answer will have no roots in it at all. Two square roots is one too many.

2:01The obvious move is to square both sides where they stand. Do that, and a cross term appears: twice the product of the two roots. A product of two roots is another root. You are no better off than when you started. So move one root across to the other side first. The left is now a single root. The right is two a, minus the other root. Square that, and the right becomes four a squared, minus four a times that root, plus whatever sits under it.

2:34Only one root survives. That is what isolating buys. Squaring a sum of two roots keeps two of them; squaring a number minus one root keeps one. The step that looks like tidying is the step the derivation cannot do without. Now clear up. On both sides the x squared, the y squared and the c squared all cancel. What is left is short: four c x equals four a squared, minus four a times the surviving root.

3:08Divide by four a, and that root equals a minus c x over a. Square once more. The left is x minus c, squared, plus y squared. The right is a squared, minus two c x, plus c squared x squared over a squared. Cancel again, gather the x terms on one side and the y term on the other. You land on x squared over a squared, plus y squared over a squared minus c squared, equals one.

3:40Two squarings, and not a root in sight. That denominator, a squared minus c squared, is correct but nameless. A standard form should not have a subtraction hiding in the floor of a fraction. And we already know what it is. The three lengths sit in a right-angled relation: a squared equals b squared plus c squared. Rearranged, a squared minus c squared is exactly b squared - half the short axis, squared.

4:14So the equation becomes x squared over a squared, plus y squared over b squared, equals one. That substitution is not cosmetic: it is where an earlier result gets spent. Without it the denominator has no name, and nothing in the equation would tell you what the number under y squared measures. The tidy form is bought with a theorem. Stop, and notice what has been shown and what has not.

4:46We started from a place on the curve and arrived at the equation. So every place of the curve satisfies the equation. Nothing yet says the reverse. For all the algebra knows, the equation might also hold at places that are nowhere near the curve. That is not a pedantic worry. Squaring is the step that loses information. It forgets signs. An equation reached by squaring twice can easily have picked up solutions the original never had.

5:19So the argument is half done, and the missing half is the interesting one. Take a solution of the equation, knowing nothing about fixed points, and show that its two distances add to two a. Here is that converse. Take x and y satisfying the equation, with the offset smaller than half the long axis. From the equation, y squared is b squared times one minus x squared over a squared.

5:51Put that into the distance to the left point: x plus c, squared, plus y squared. Replace b squared by a squared minus c squared, and expand. What comes out is a squared, plus two c x, plus c squared x squared over a squared. Look at that. It is a perfect square: a plus c x over a, all squared. So that distance is a plus c x over a.

6:19One caution the tidy answer hides. Pulling a square out of a root gives you a size, not an expression, unless the expression is not negative. Here it never is, because x never exceeds a and the offset is smaller than a. Over twenty-five exact places of the curve, the number of times that expression came out negative is zero. The other distance goes the same way, with one sign changed.

6:50The distance to the right point is a minus c x over a. Add them. The c x terms cancel, and the sum is two a. So the place satisfies the definition, and the converse is done. Both directions hold now, which means the equation and the curve are the same set of places. That is worth checking rather than believing. Over a window of two thousand nine hundred and eleven places, the set holds at twelve of them, and the equation holds at twelve, and they are the same twelve.

7:26Places the set holds that the equation misses: zero. Places the equation holds that the set does not: zero. An equation that forgot the offset, with the same number under both squares, gets twenty-two of those places wrong. Now look at the shape of those two answers, because this is the best idea in the whole topic. a plus c x over a. a minus c x over a. Both of them are straight lines in x. Not nearly straight. Exactly.

8:02Slide the place to the right, and one distance grows at a steady rate while the other shrinks at exactly the same rate. That rate is c over a: the offset divided by half the long axis. So the sum cannot move. The constant sum is not a coincidence the algebra confirms afterwards. It is visible in the shape of the answer. Measure it rather than say it. Twenty-five exact places make three hundred pairs, and two hundred and eighty-eight of them sit at different x.

8:38Across all of them, the number of distinct rates the far distance changes at is one. For the near distance, also one. The two rates are four fifths and minus four fifths, and they add to nothing. The equation also says where the curve cannot go. x squared over a squared equals one minus y squared over b squared, and a square is never negative, so that is at most one.

9:09Therefore x squared is at most a squared: x lies between minus a and a. Run the same argument the other way round for y. The curve is trapped inside a rectangle - inclusively, with the ends belonging to it. Walk the flat axis of a set with its fixed points at minus four and plus four and its constant ten, and you meet it at exactly two places: minus five and plus five.

9:38Walk the upright axis and you meet it twice more, at minus three and plus three. Those are the four places where the curve touches its box - and notice that the two half-lengths, five and three, were found by walking, not read off the equation. The four corners are a different story. None of them is on the curve, and none of them is inside it. With the equation trusted, it can be read.

10:08Take x squared over twenty-five, plus y squared over nine, equals one. Twenty-five is the bigger number and it sits under x, so the long axis lies flat. Then a is five, b is three, and the offset is the root of twenty-five minus nine, which is four. Fixed points at plus and minus four along the flat axis. Ends of the long axis at plus and minus five. The long axis measures ten, the short one six, and the ratio of offset to half the long axis is four fifths.

10:47Here is the part that matters. The bigger denominator naming the long axis is not a convention someone chose. It is a consequence. Half the long axis was already proved bigger than half the short one, so whichever denominator is larger has to be the long one, squared. A rule with a reason survives when a later curve breaks it. Now the case that gets misremembered more than any other.

11:18Put the two fixed points on the upright axis instead, at zero and plus or minus four, with the same constant ten. The equation is x squared over nine, plus y squared over twenty-five, equals one. The twenty-five has moved. It sits under y now, and the long axis stands upright. Scored over the same window, that set and that equation disagree at zero places. The habit worth breaking is the one that says the big number goes under x.

11:53It does not. The big number goes under whichever variable the long axis belongs to. Read nine equations of this kind and five of them stand upright while four lie flat, and not one names neither. If you had assumed flat every time, you would have been wrong five times out of nine. One more trap, and this one is purely mechanical. Take nine x squared plus four y squared equals thirty-six.

12:23Read the coefficients where they stand: nine beats four, so you would call the long axis flat. That is wrong, because a standard form has one on the right-hand side. Divide everything by thirty-six. You get x squared over four, plus y squared over nine, equals one. Now nine sits under y, and the long axis stands upright. The denominators are the reciprocals of the coefficients, so dividing turns the order of size upside down.

12:57Always divide first. And if there is nothing on the right, or one of the two squares is missing, there is no curve of this kind to be had - two different complaints, and worth saying which one you are making. Going the other way: given two places, find the curve. Suppose the long axis lies flat, and the curve passes through four, three and through minus one, four. Substituting gives sixteen over a squared plus nine over b squared equals one, and one over a squared plus sixteen over b squared equals one.

13:38Those two are not linear in a squared and b squared. But they are perfectly linear in the reciprocals. Call one over a squared u, call one over b squared v, and the pair solves at once. a squared comes out two hundred and forty-seven sevenths, and b squared two hundred and forty-seven fifteenths. Clear the fractions and the equation is seven x squared plus fifteen y squared equals two hundred and forty-seven.

14:08One check is left. Is a squared bigger than b squared? It is - so the flat long axis the question stipulated is what actually came out. Two places do not pin down a curve on their own; the stipulation is doing work too. Two places these curves turn up. Lean a rod fifteen units long with one end on each axis, and mark a place six units along it.

14:37As the rod slides, that mark traces x squared over eighty-one plus y squared over thirty-six equals one. Over twenty-three positions of the rod, the number of marked places that miss that curve is zero. Move the mark to five fifteenths along instead, and all twenty-three miss it. They are tracing a different curve. Which curve you get is settled by where on the rod you stand. Second, an arch shaped like half of one of these, eight units wide and two tall.

15:11Two and a half units in from the middle, its squared height is thirty-nine sixteenths - which is not the square of any fraction, so the honest answer is not a tidy one. At the middle the arch stands two units tall; at either end, nothing. Two frame decisions, one theorem spent, and a converse whose answer turns out to be a straight line. That is the whole of it.