Tangents, Normals and Chords

Transcript

0:00Introduction

Hello everyone! First, a small puzzle. Take a hyperbola with its two asymptotes, the lines its branches come closer and closer to, but never meet. Now hold a ruler parallel to one asymptote, and slide it across. Will it ever just touch the hyperbola? Keep your answer ready. Today's formulas look like the ellipse ones with one sign changed, and that sign is where students lose marks. They come up often in JEE papers, so let's begin.

0:30Tangent at a point

Here is our hyperbola: x squared over a squared, minus y squared over b squared, equals one. The small two up there means squared, and the small ones below are subscripts. The tangent at x one, y one is x times x one over a squared, minus y times y one over b squared, equals one. Same pattern as the ellipse, and the minus sign stays. Okay? Try one from the course: the tangent to x squared over nine, minus y squared over four, equals one, at three root two comma two. The tick-shaped sign is the square root. We get three root two times x over nine, minus two y over four, equals one. Multiply by six, and you get two root two times x, minus three y, equals six. Clear? With the parameter theta, the Greek letter here, the point is a times sec theta, comma, b times tan theta. The tangent becomes x sec theta over a, minus y tan theta over b, equals one. With me?

1:31Tangents of slope m

On the right: the two branches, the foci S and S prime, the little dash read prime, the red dashed directrices, x equals plus or minus a over e, with the right one labelled x equals a over e, and a point P joined to both foci. Given only the slope m, the tangents are y equals m x, plus or minus the square root of, a squared times m squared, minus b squared. So tell me, does every slope give a tangent now? Not every slope, and here is why: the quantity under the root must be positive, so a squared times m squared must be more than b squared. That means mod m, with the two vertical bars, is greater than b over a. The dashed grey lines are the asymptotes, y equals plus or minus b over a times x, so a tangent must be steeper than them. Got it? So the line y equals m x plus c touches the hyperbola when c squared equals a squared times m squared, minus b squared. This is the condition of tangency. Clear? And the point of contact is minus a squared times m, over c, comma, minus b squared over c, in the big round brackets. Both coordinates have a minus sign here. Okay? Now our puzzle. The ruler parallel to an asymptote has slope plus or minus b over a. Then c squared is zero, so c is zero, and the line is the asymptote itself, which never meets the hyperbola. So no tangent is parallel to an asymptote. Puzzle solved, right?

3:07Normals

Next, the normal, perpendicular to the tangent. At x one, y one, it is a squared times x over x one, plus b squared times y over y one, equals a squared plus b squared. Watch the signs: plus, and plus, where the ellipse had minus signs. Okay? From the course: the normal to x squared over sixteen, minus y squared over nine, equals one, at five comma nine over four. Sixteen plus nine gives twenty five, so sixteen x over five, plus nine y divided by nine quarters, equals twenty five. That second term is just four y. Multiply by five, and you get sixteen x plus twenty y, equals one hundred and twenty five. Right?

3:53The director circle

When two tangents are at right angles, they meet on the circle x squared plus y squared equals a squared minus b squared. This is the director circle; for the ellipse it had a plus sign. Okay? From the course: x squared over twenty five, minus y squared over sixteen, equals one. Twenty five minus sixteen gives nine, so the director circle is x squared plus y squared equals nine, radius three. Easy, right? Now the same numbers, swapped: x squared over sixteen, minus y squared over twenty five, equals one. What is its director circle? Sixteen minus twenty five gives minus nine. But wait, x squared plus y squared can never be minus nine! Here a is less than b, so there is no director circle at all. Got it?

4:46Chords: chord of contact T = 0 and mid-point chord T = S1

Capital S, with the three-line sign meaning is defined as, is x squared over a squared, minus y squared over b squared, minus one, and S one is its value at x one, y one. Capital T is x times x one over a squared, minus y times y one over b squared, minus one. Okay? From x one, y one, draw the two tangents to the hyperbola. The chord joining their points of contact is the chord of contact: T equals zero. Right? And the chord whose mid-point is x one, y one is T equals S one. Chord of contact, T equals zero; mid-point given, T equals S one. Clear?

5:28Example 1: tangents of slope 2 to x2/16 - y2/9 = 1

Now Example one from the course. Find the tangents of slope two to x squared over sixteen, minus y squared over nine, equals one. But first, check whether slope two is even allowed. Pause the video and try it first. Done? Okay, let's check. Here a squared is sixteen, b squared is nine, and m is two. So b over a is three quarters, and two is bigger, so yes, the tangents exist. With me? Now c squared, a squared times m squared minus b squared, is sixteen times four, minus nine, which is fifty five; the dot means times. Oh no, careful if you wrote sixty four plus nine, because that is the ellipse formula. Got it? So c is plus or minus root fifty five, and the tangents are y equals two x, plus or minus root fifty five. So we get two parallel tangents. Clear?

6:27Example 2: chord of x2/4 - y2/9 = 1 bisected at (4, 3)

Now a harder one, Example two from the course. Find the chord of x squared over four, minus y squared over nine, equals one, bisected at four comma three. Here is a hint: bisected means four comma three is the mid-point. Pause the video and try it first. Done? Okay, let's check. Many students put four comma three into the tangent-style formula, and get x minus y over three, equals one. Is that the chord? No, no, wait, because that is T equals zero, the chord of contact formula! Multiply by three, and it is three x minus y, equals three. Put in four comma three, and twelve minus three gives nine, not three. So that line misses its own mid-point. Right? Mid-point given, so we need T equals S one, at four comma three. So x minus y over three, minus one, equals sixteen over four, minus nine over nine, minus one. With me? The minus ones cancel, so x minus y over three equals four minus one, which is three. Multiply by three, and we get three x minus y, equals nine. And three times four, minus three, gives nine, so it passes through four comma three. That's it, we are done!

7:48Quick revision: tangent, normal, director circle, chords

So, quick revision. The tangent is x times x one over a squared, minus y times y one over b squared, equals one, and y equals m x plus c touches when c squared equals a squared times m squared, minus b squared. The normal has plus signs, and the director circle is x squared plus y squared equals a squared minus b squared. Chord of contact, T equals zero; mid-point given, T equals S one. Now try the practice problems on this page. All the best!