Confusing Chord of Contact proof by Cambridge (1 Viewer)

[Chris100]

That leap in the proof between the pages (...and since P lies on the tangent at B....But the first identity shows that A..)
I don't understand how the first identity (and i dont even know which identity it's talking about) shows that A lies on xx0=2a(y+y0).
Please clarify this proof for me

 

[QZP]

I'll try to rephrase it:

Suppose you have two points A(x1, y1), B(x2, y2) on the parabola x^2 = 4ay whose tangents pass through a common point P(x0, y0) outside the parabola. We now want to find the equation for the chord of contact (line AB).

Subbing P into the tangents at A and B, shows that:
"Identity 1" (this is what they were referring to): x0.x1 = 2a (y0 + y1)
"Identity 2": x0.x2 = 2a (y0 + y2)

Now take a step back and notice that both A, B both satisfy the equation x0.x = 2a(y0 + y).
Hence, this must be the equation for the CHORD OF CONTACT because there can only be ONE possible line that passes through both A and B.

 

[enigma_1]

You probably wont have to reproduce this particular proof in an exam. You will learn the formula off by heart after doing a bunch of exercises. You can just rote learn it for an exam, but yeah it's good to know where it comes from sorta

 

[Chris100]

Ah I see thank you!
Also, is this formula allowed to be used straight away (without derivation) similar to the sum of roots formula and quotient rule, or do you have to derive it every time similar to what we do with the parametric equation of tangents?


Edit: just saw hyper's post as the thread updated as I posted the above reply, so disregard

 

[panda15]

You probably wont have to reproduce this particular proof in an exam. You will learn the formula off by heart after doing a bunch of exercises. You can just rote learn it for an exam, but yeah it's good to know where it comes from sorta

I would definitely advise against rote learning the formula. 99 times out of 100, if a question involves a parametric equation, it will get you to prove the equation first.

 

[enigma_1]

The question may guide you into proving the formula, like you wont have to do it yourself. I've never had to randomly prove it. But yeah the question will mostly guide you with it. It's a retarded proof the one they use 'elegantly' or whatever. There are alternatives.

 

[enigma_1]

Or they'll tell you "Using the formula ____________ (for chord of contact), do this ___"

 

[panda15]

The question may guide you into proving the formula, like you wont have to do it yourself. I've never had to randomly prove it. But yeah the question will mostly guide you with it. It's a retarded proof the one they use 'elegantly' or whatever. There are alternatives.

I haven't seen a HSC paper where they guide you through a proof in a veeeeery long time, and now with MC, I don't think you will see one again because they don't have as many marks to allocate to long response questions. Plus it's extension 1, they're not gonna hold your hand through a proof that you should know.

Either that, or like you said, they will give you the equation straight up.

 

[braintic]

I haven't seen a HSC paper where they guide you through a proof in a veeeeery long time, and now with MC, I don't think you will see one again because they don't have as many marks to allocate to long response questions. Plus it's extension 1, they're not gonna hold your hand through a proof that you should know.

Either that, or like you said, they will give you the equation straight up.

Can you tell me one HSC paper that has a chord of contact question. I've looked all the way back to the 70s, and I haven't found one question.
Chord of Contact is actually not mentioned in the syllabus at all - all it says for parameters is "Applications to problems concerned with tangents, normals and other geometric properties."

So I believe that someone probably included this in their textbook years ago just for the hell of it, and all of us have been teaching this every year thinking it could be examined, while the Board has been letting us teach material from outside the syllabus without saying a thing.

 

[QZP]

^ Wow that is so true lol.

 

[Chris100]

Sick of your shit BOS

 

[panda15]

Can you tell me one HSC paper that has a chord of contact question. I've looked all the way back to the 70s, and I haven't found one question.
Chord of Contact is actually not mentioned in the syllabus at all - all it says for parameters is "Applications to problems concerned with tangents, normals and other geometric properties."

So I believe that someone probably included this in their textbook years ago just for the hell of it, and all of us have been teaching this every year thinking it could be examined, while the Board has been letting us teach material from outside the syllabus without saying a thing.

Umm, chord of contact is in the syllabus. Straight from the syllabus on BOS website under 9.6:

The equation of the chord of contact of the tangents from an external point.

And so who cares if it hasn't been asked. It's bound to come around eventually. And going by the last 10 years of HSC papers, they're not gonna hold your hand through the proof. I'm saying there's no point of rote learning the formula. They will either give it to you, or you will be asked to prove it. You're much better off remembering how to derive the formulas rather than simply rote learning them.