Math Is Fun Forum

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

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Represent the following equation as a hyperbola

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'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'
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gAr

Member

Registered: 2011-01-09

Posts: 3,482

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.

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"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?

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

scientia

Member

Registered: 2009-11-13

Posts: 224

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.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Represent the following equation as a hyperbola

Ok thanks

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'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'
I'm not crazy, my mother had me tested.