Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2013-01-16 02:36:38

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 14,207
Website

Represent the following equation as a hyperbola


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

Offline

#2 2013-01-16 03:40:12

gAr
Member
Registered: 2011-01-09
Posts: 3,462

Re: Represent the following equation as a hyperbola

Hi Agnishom,

You mean as in the one given here:  wiki?
Put h = k = m = 1, and you have this equation.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

Offline

#3 2013-01-16 03:57:58

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 81,413

Re: Represent the following equation as a hyperbola

Hi;

He can put it in standard form with a rotation of 45 degrees clockwise and a translation by the √2

Just involves the substitutions
of

and then replacing x1 by x1+ √2.

You might download this

http://math.sci.ccny.cuny.edu/document/show/2685

rename the file to Rotation of Axes.pdf This will explain some of this, won't make you as good as scientia or bob bundy with these transformation problems but it is a start.

View Image: first.gif View Image: second.gif

In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

Offline

#4 2013-01-16 04:23:29

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,092

Re: Represent the following equation as a hyperbola

hi bobbym

Like the light sabre by the way.  You beat me to it. 

Agnishom: Here's my version:

Substitute* x = X +1  and  y = Y + 1, where X and Y are new variables.

So we now have a more familiar XY = 0 (the rectangular hyperbola)

Now substitute* X = x/a - y/b    and Y = x/a + y/b

*  substitutions like these preserve the hyperbolic nature of the curve.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

#5 2013-01-16 06:50:47

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: Represent the following equation as a hyperbola


In questions of this sort I tend look for a transformation to get rid of the
term:

When this is substituted into the original equation, the term in

is

We want this to vanish, so any

such that
and
will do. So we take
. Hence

Thus under the transformation the curve

becomes the hyperbola
. Furthermore as the transformation

represents a clockwise rotation of 45° about the origin followed by an enlargement of

at the origin, the conic section is preserved, i.e. the original curve is indeed a hyperbola.

NB: Be careful when using linear transformations on curves: only rotations, reflections and enlargements/contractions by a nonzero factor preserve conic sections. Any other transformation may distort the curve and alter its original nature.

Offline

#6 2013-01-16 13:59:21

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 14,207
Website

Re: Represent the following equation as a hyperbola

Ok thanks


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Humanity is still kept intact. It remains within.' -Alokananda

Offline

Board footer

Powered by FluxBB