Definition and Standard Form

Transcript

0:00Introduction

Hello everyone! Let us start with a small experiment. Fix two pins on a board, and tie the two ends of a thread to them, with the thread longer than the gap between the pins. Pull the thread tight with a pencil, and move it all the way round. What curve does the pencil draw? And how do we write its equation? Ellipse questions come up often in JEE papers, so let's begin.

0:27Two ways to define an ellipse

So, the pencil draws an ellipse! The first definition: the locus of a point P with P S equal to e times P M; the small dot means times. Here e is the eccentricity, and the two less than signs say it is more than zero and less than one. Okay? S is a fixed point, the focus, and P M is the distance of P from a fixed line, the directrix. The two dashed red lines in the figure are directrices. Right? Now the second definition, the one our pencil used. Take two fixed points, S and S prime; the little dash is read as prime. These are our two pins. The orange lines are S P and S prime P, and their sum stays equal to two a, the length of the thread. Clear? Now let me ask you this: what happens if the gap between the pins is two a, or even more? If the gap equals two a, the tight thread lies flat on the segment between the pins; if the gap is bigger, it cannot even reach. No ellipse either way, so the definition needs S S prime to be less than two a. Got it?

1:31The standard form

Now the standard form, with the centre at the origin. The course writes x squared over a squared, plus y squared over b squared, equals one; the small two means squared. And b squared equals a squared, times one minus e squared. With me? Now tell me, since e lies between zero and one, which is bigger, a or b? Yes, exactly! One minus e squared is less than one, so b squared is less than a squared. The arrow means implies: a is greater than b, which is greater than zero. Okay? The foci are plus or minus a times e, comma zero; that is the plus-minus sign. So the gap between S and S prime is two times a times e. Right? And the directrices are x equals plus or minus a over e. Focus, a times e; directrix, a over e. Never mix these two up. Clear? The major axis has length two a, along the x axis, and the minor axis has length two b; in the figure, a and b mark half of each.

2:34Latus rectum and focal distances

Okay? Next, the latus rectum. Its length is two b squared over a, and its ends are plus or minus a times e, comma, plus or minus b squared over a. See the x coordinate of the ends, plus or minus a times e, the same as the foci. Got it? Now the focal distances of a point P, x comma y, on the ellipse. S P equals a, minus e times x, and S prime P equals a, plus e times x. No square roots at all! Right? Add them, and the e times x terms cancel: S P plus S prime P equals two a, for every point. That is our thread from the start: this equation describes exactly the curve the pencil drew. With me?

3:16Eccentric angle and auxiliary circle

Now look at the dashed grey circle in the figure. It is the auxiliary circle, x squared plus y squared equals a squared, the circle with the major axis as its diameter. Okay? Every point of the ellipse can be written as a times cosine theta, comma, b times sine theta, where theta, the Greek letter, is called the eccentric angle.

3:36When b is greater than a: major axis along y

Clear? Now, what if b is bigger than a? Then the major axis is along the y axis, and the roles swap: a squared equals b squared, times one minus e squared. The foci move to zero comma plus or minus b e, and the latus rectum becomes two a squared over b. Okay? So here is a topper's trick: before any formula, compare the two denominators. The bigger one decides the major axis. Clear? Test it on one from the course: nine x squared plus four y squared equals thirty six. Divide by thirty six: x squared over four, plus y squared over nine, equals one. Nine is bigger, so b is bigger. Now, what is e? Oh no, careful, do not write one minus nine over four, because that is negative! Here a squared is four, so e squared is one minus four over nine, which is five over nine. So e is root five over three; that is the square root sign. Then b times e is root five, so the foci are zero comma plus or minus root five, and the latus rectum is eight over three. Got it?

4:40Example 1: eccentricity, foci, directrices and latus rectum

Okay, Example one, from the course. Find the eccentricity, the foci, the directrices and the latus rectum of x squared over twenty five, plus y squared over nine, equals one. Pause the video, and try it first. Done? Okay, let's check. Twenty five is the bigger denominator, so a squared is twenty five, a is five, and b squared is nine. Then e squared is one minus nine over twenty five, which is sixteen over twenty five, so e is four over five. Okay? The foci are plus or minus a times e, comma zero. Five times four over five is four, so the foci are plus or minus four, comma zero. Right? Now the directrices. Five and four over five again, so x equals plus or minus four, isn't it? No, no, wait, that was a times e, the focus! The directrix uses a over e: five, over four fifths, gives twenty five over four. So the directrices are x equals plus or minus twenty five over four. Clear? And the latus rectum is two b squared over a: two times nine over five, which is eighteen over five. All four answers are done!

5:51Example 2: ellipse from focus (1, 0) and directrix x = 4

Now a harder one, Example two from the course. An ellipse centred at O, the origin, has a focus at one comma zero, and the corresponding directrix is x equals four. Find it. Hint: write a times e and a over e. Pause the video and try it first. Done? Okay, let's check. The directrix is at four, so a is four, right? No, no, careful, four is not the value of a here! The focus is a times e, comma zero, so a times e is one. And the directrix is x equals a over e, so a over e is four. Okay? Now a lovely trick: multiply these two. The e cancels, and a squared is one times four, which is four. With me? Next, b squared is a squared, times one minus e squared, which is a squared minus, a times e, the whole squared. That is four minus one, which is three. Right? So the ellipse is x squared over four, plus y squared over three, equals one. And four is bigger than three, so the major axis is along x, exactly where the focus sits. If you got this one on your own, very well done!

7:04Quick revision: definitions, foci, directrices, major axis

So, quick revision. An ellipse: S P plus S prime P equals two a, or P S equals e times P M, with e between zero and one. Standard form: b squared is a squared times one minus e squared, foci plus or minus a times e, directrices plus or minus a over e. And compare the denominators first; the bigger one gives the major axis. Now go and try the practice problems on this page. All the best!