Tangents, Normals and Chords

Transcript

0:00Introduction

Hello everyone! Let us start with a small puzzle. Take the parabola y squared equals sixteen x, and a line of slope two. Slide that line up and down. In exactly one position, it just touches the curve. Where does that line cut the y axis? Keep your guess; you will answer it in one line by the end. Tangents, normals and chords of the parabola come up often in JEE papers, so let's begin.

0:27Tangents to y²=4ax

So, our parabola is y squared equals four a x; the small two means squared. At a point x one, y one, where the small ones below are subscripts, the tangent is y times y one equals two a times, x plus x one. Okay? Here is one from the course. What is the tangent to y squared equals eight x, at the point two comma four? Four a is eight, so a is two, and two a is four. So four y equals four times, x plus two, which is y equals x plus two. Many students write eight there; careful, the formula has two a, not four a, in front. Right? Now take the parameter point t, that is a times t squared, comma, two a times t. The tangent there is t times y equals x plus a times t squared. With me? And when the slope m is given, with m not equal to zero, the crossed equals sign, the tangent is y equals m x plus a over m. It touches the curve at the point a over m squared, comma, two a over m, in big round brackets. Clear? Now take any line y equals m x plus c, where c is the intercept on the y axis. When does it just touch the parabola? Yes, exactly, when c equals a over m, and only then; i f f on screen is short for if and only if. This is the tangency condition, and it cracks our opening puzzle. Got it?

2:06Normals to y²=4ax

Now the normal: the line through the point of contact, perpendicular to the tangent. At t, it is y equals minus t times x, plus two a times t, plus a times t cubed; the small three means cubed. Try this one from the course: the normal to y squared equals twelve x at three comma six. Four a is twelve, so a is three. Then two a times t is six, so t is one. So y equals minus x, plus six, plus three, which is x plus y equals nine. Okay? And when t is not zero, this normal cuts the parabola once more, at the point with parameter t two, which equals minus t, minus two over t. Clear?

2:53Chords and focal chords

Next, the chord joining the points t one and t two is two x, minus, t one plus t two, times y, plus two a times t one times t two, equals zero. Now let it pass through the focus, a comma zero. What must t one times t two be? Put x equals a and y equals zero: two a, plus two a times t one times t two, equals zero. The arrow means implies, so t one times t two equals minus one. So if one end is t one, the other end is t two equals minus one over t one. Got it? And the focal chord at t has length a times, t plus one over t, the whole squared. Keep this ready for Example two. Right?

3:41Where two tangents meet

Now the tangents at t one and t two meet at the point a times t one times t two, comma, a times, t one plus t two. If t one and t two are the ends of a focal chord, where does this point lie? Yes, exactly, t one times t two is minus one, so the x coordinate becomes minus a, and the point sits on the directrix! With me? The tangent at t has slope one over t, but only when t is not zero; at t equals zero the tangent is the y axis, a vertical line with no slope. Here t one times t two is minus one, so neither of them is zero, and both slopes exist. Their product is one over t one times t two; the dot means times. That is minus one, so these tangents are perpendicular as well. Okay? And the other way round too: any two perpendicular tangents of this parabola meet on the directrix. Clear?

4:41Two short names: T and S₁

Now two short names. Capital T, with the three-line sign meaning is defined as, stands for y times y one, minus two a times, x plus x one. And S one is y one squared, minus four a times x one. Right? From a point x one, y one outside the parabola, draw the two tangents to it. The chord joining their points of contact is the chord of contact, and its equation is T equals zero. Got it? And the chord whose mid-point is x one, y one is T equals S one. Please do not mix these two up. Okay?

5:19Example 1: tangent of slope 2 to y2 = 16x, point of contact

Now, remember our opening puzzle? Here it is as Example one, from the course. Find the tangent of slope two to y squared equals sixteen x, and its point of contact. Pause the video and try it first. Done? Okay, let's check. The slope m is two, and c is a over m. So sixteen over two, which gives eight, isn't it? No, no, wait, sixteen is four times a, not a by itself! Compare with y squared equals four a x: four a is sixteen, so a is four, and m is two. Right? So c is four over two, which is two, and the tangent is y equals two x plus two. So the special line cuts the y axis at two, and the puzzle is solved. Clear? And the point of contact is a over m squared, comma, two a over m. That is four over four, comma, eight over two, which is one comma four. Got it?

6:21Example 2: other end and length of a focal chord of y2 = 4x

Now a harder one, Example two from the course. The point four comma four is one end of a focal chord of y squared equals four x. Find the other end, and the length of the chord. Here is a hint: find t first. Pause the video and try it. Done? Okay, let's check. Four a is four, so a is one, and the focus is one comma zero. Then t squared, comma, two t is four comma four, so t is two. With me? Now the other end. Many students say, by symmetry, it must be four comma minus four. Is that right? Oh no, careful, the chord from four comma four to four comma minus four is the line x equals four, and it misses the focus, one comma zero. So that is not a focal chord at all. Right? For a focal chord, the two parameters multiply to minus one; the little dash is read t prime, and it marks the other end. So t prime is minus one over t, which is minus one half. The other end, a times t prime squared, comma, two a times t prime, is one quarter, comma, minus one. Clear? And the length is a times, t plus one over t, the whole squared. Two plus one half is five over two, and its square is twenty five over four, and we are done.

7:44Quick revision: tangent, normal and focal chord formulas

So, quick revision. Tangent at x one, y one: y times y one equals two a times, x plus x one; and y equals m x plus c touches when c equals a over m. Normal at t: y equals minus t times x, plus two a times t, plus a times t cubed; and a focal chord has t one times t two equal to minus one. Tangents at the ends of a focal chord are perpendicular, and meet on x equals minus a, the directrix. You caught two real traps today, very well done. Now try the practice problems on this page. All the best!