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 Post subject: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 07:06:46 UTC
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Attempt at a solution:

i) circle has equation (x+c)^2 + (y-c)^2 = (2c)^2
hyperbola has equation y=1/x
Substituting y=1/x into the equation for the circle gives the equation x^4 + 2cx^3 -2cx + 1 = 0

ii)The equation has roots x=alpha, alpha, beta and beta because this is where the two curves touch. It is a quartic equation therefore the two points of contact must be double roots.

iii) Not sure how to do part 3 - help would be greatly appreciated!

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 Post subject: Re: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 07:14:49 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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simonsmith wrote:

Attempt at a solution:

i) circle has equation (x+c)^2 + (y-c)^2 = (2c)^2
hyperbola has equation y=1/x
Substituting y=1/x into the equation for the circle gives the equation x^4 + 2cx^3 -2cx + 1 = 0

ii)The equation has roots x=alpha, alpha, beta and beta because this is where the two curves touch. It is a quartic equation therefore the two points of contact must be double roots.

iii) Not sure how to do part 3 - help would be greatly appreciated!

What image?

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 Post subject: Re: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 07:20:27 UTC
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simonsmith wrote:

Attempt at a solution:

i) circle has equation (x+c)^2 + (y-c)^2 = (2c)^2
hyperbola has equation y=1/x
Substituting y=1/x into the equation for the circle gives the equation x^4 + 2cx^3 -2cx + 1 = 0

ii)The equation has roots x=alpha, alpha, beta and beta because this is where the two curves touch. It is a quartic equation therefore the two points of contact must be double roots.

iii) Not sure how to do part 3 - help would be greatly appreciated!

Start by writing down the Viete's formulas.

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 Post subject: Re: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 07:22:10 UTC
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There's a name for a result that obvious?

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 Post subject: Re: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 10:53:55 UTC
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There's a name for a result that obvious?

I usually refuse to call it Viete's formula, but only as "symmetry of roots". However, that is the name given in the wiki article.

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 Post subject: Re: confusing polynomial question - stuck on part 3Posted: Fri, 13 Jan 2012 20:43:48 UTC
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outermeasure wrote:
There's a name for a result that obvious?

I usually refuse to call it Viete's formula, but only as "symmetry of roots". However, that is the name given in the wiki article.

OK, that's fair then.

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