Rectangular Hyperbola Help? (1 Viewer)

[skrye4]

Hey Everyone,

I just took a look at the Maths E2 syllabus, and one dot point which kind of confused me was:

"The student is able to prove that the hyperbola with equation xy = 1/2(a²) is the hyperbola x² - y² = a² referred to different axes."

Does anyone reckon they might be able to explain this to me? How would you go about proving that?

Thanks in advance for your help

 

[Sy123]

Hey Everyone,

I just took a look at the Maths E2 syllabus, and one dot point which kind of confused me was:

"The student is able to prove that the hyperbola with equation xy = 1/2(a²) is the hyperbola x² - y² = a² referred to different axes."

Does anyone reckon they might be able to explain this to me? How would you go about proving that?

Thanks in advance for your help

It basically means that you are expect to be able to prove that if you rotate the curve x^2-y^2 = a^2 45 degrees you get the hyperbola xy= 1/2 a^2
So referring to different axis means to rotate it basically.

The common proof of this is to take the parametric equations of the hyperbola (asec(t), btan(t)), and make a complex number representing this locus



On the arg-and diagram the locus of this is a hyperbola by definition, then rotate this complex number 45 degrees anti-clockwise, this is done by multiplying both sides by 45 degrees.
You then get a complex number of the form

THis means the new locus is (x(t),y(t)), one then needs to prove that the new locus satisfies xy = 1/2 a^2

Of course there are many other methods if you want to be creative about it. Such as using the definition of a rectangular hyperbola in its conical definition (ratio between a fixed point and fixed line), taking these fixed lines and focii, and rotating them about the origin using trigonometry.

 

[braintic]

See attachment: View attachment Rotated Hyperbola.pdf

 

[obliviousninja]

See attachment: View attachment 28570

Is knowing how to prove the directrices, focii, and vertices required?

 

[braintic]

Is knowing how to prove the directrices, focii, and vertices required?

I've never seen this rotation concept tested in the HSC with the hyperbola, let alone the foci and directrices.
So ... technically yes ... in reality not likely.

 

[asianese]

There's a (significant) leap in the proof - how do you just ;know; to go XY = .... ? It's not likely to be tested, but you know the general concepts of the proof. If ever they'd ask for the proof, they'd probably split into 2 parts and then make you deduce it.

 

[braintic]

There's a (significant) leap in the proof - how do you just ;know; to go XY = .... ? It's not likely to be tested, but you know the general concepts of the proof. If ever they'd ask for the proof, they'd probably split into 2 parts and then make you deduce it.

You 'know' based on the previous line - "given that ..."
You know you need to bring an x^2 - y^2 into the mix in order to eliminate x and y.
Once you realise that, I think multiplying X by Y is pretty obvious.

 

[Fred2013]

Hey Everyone,

I just took a look at the Maths E2 syllabus, and one dot point which kind of confused me was:

"The student is able to prove that the hyperbola with equation xy = 1/2(a²) is the hyperbola x² - y² = a² referred to different axes."

Does anyone reckon they might be able to explain this to me? How would you go about proving that?

Thanks in advance for your help

Look in Terry Lee's book or Cambridge 4u. They explain it pretty well there