Fdtxjr

When did F become S? Why?

Did the new image of black hole confirm the general theory of relativity?

Can the Right Ascension and Argument of Perigee of a spacecraft's orbit keep varying by themselves with time?

Does Parliament need to approve the new Brexit delay to 31 October 2019?

How to support a colleague who finds meetings extremely tiring?

What do I do when my TA workload is more than expected?

Was credit for the black hole image misappropriated?

"... to apply for a visa" or "... and applied for a visa"?

Does Parliament hold absolute power in the UK?

Example of compact Riemannian manifold with only one geodesic.

Loose spokes after only a few rides

What's the point in a preamp?

What information about me do stores get via my credit card?

Is it ok to offer lower paid work as a trial period before negotiating for a full-time job?

How do spell lists change if the party levels up without taking a long rest?

Are spiders unable to hurt humans, especially very small spiders?

How do I design a circuit to convert a 100 mV and 50 Hz sine wave to a square wave?

For what reasons would an animal species NOT cross a *horizontal* land bridge?

Is there a way to generate uniformly distributed points on a sphere from a fixed amount of random real numbers per point?

The following signatures were invalid: EXPKEYSIG 1397BC53640DB551

Homework question about an engine pulling a train

How did the crowd guess the pentatonic scale in Bobby McFerrin's presentation?

What to do when moving next to a bird sanctuary with a loosely-domesticated cat?

How do you keep chess fun when your opponent constantly beats you?

Intersecting Integral points $(x,y,z)$ of $3x+5y+4z=45$ and $z^2+xy=15$?

0

$begingroup$

I am trying to find all the intersecting integral points $(x,y,z)$ of the plane $$3x+5y+4z=45$$ and the one-sheeted paraboloid $$z^2+xz=15$$.

I noticed that
$$x=4t-(9-y)$$
$$z=-3t+2(9-y)$$
So I replaced $x,z$. I ended up with the hyperbola $$9 t^2 + 16 t y - 108 t + 5 y^2 - 81 y + 309 = 0$$
which I solved to find infinitely many integral solutions including $$(2,7,1)$$.
My question is: Is there any easier method to find all those points knowing a common intersecting integral point?

diophantine-equations

share|cite|improve this question

edited Mar 25 at 14:53

NumThcurious

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The equation of hyperbola you got, put it in standard form and then you can find all rational (integral) solutions by using this parametrization of hyperbola.
  $endgroup$
  – ersh
  Mar 24 at 16:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your input. I’ll try it.
  $endgroup$
  – NumThcurious
  Mar 24 at 17:12
  

  
  
  
  

add a comment | 

0

$begingroup$

I am trying to find all the intersecting integral points $(x,y,z)$ of the plane $$3x+5y+4z=45$$ and the one-sheeted paraboloid $$z^2+xz=15$$.

I noticed that
$$x=4t-(9-y)$$
$$z=-3t+2(9-y)$$
So I replaced $x,z$. I ended up with the hyperbola $$9 t^2 + 16 t y - 108 t + 5 y^2 - 81 y + 309 = 0$$
which I solved to find infinitely many integral solutions including $$(2,7,1)$$.
My question is: Is there any easier method to find all those points knowing a common intersecting integral point?

diophantine-equations

share|cite|improve this question

edited Mar 25 at 14:53

NumThcurious

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The equation of hyperbola you got, put it in standard form and then you can find all rational (integral) solutions by using this parametrization of hyperbola.
  $endgroup$
  – ersh
  Mar 24 at 16:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your input. I’ll try it.
  $endgroup$
  – NumThcurious
  Mar 24 at 17:12
  

  
  
  
  

add a comment | 

$begingroup$

I am trying to find all the intersecting integral points $(x,y,z)$ of the plane $$3x+5y+4z=45$$ and the one-sheeted paraboloid $$z^2+xz=15$$.

I noticed that
$$x=4t-(9-y)$$
$$z=-3t+2(9-y)$$
So I replaced $x,z$. I ended up with the hyperbola $$9 t^2 + 16 t y - 108 t + 5 y^2 - 81 y + 309 = 0$$
which I solved to find infinitely many integral solutions including $$(2,7,1)$$.
My question is: Is there any easier method to find all those points knowing a common intersecting integral point?

diophantine-equations

share|cite|improve this question

edited Mar 25 at 14:53

NumThcurious

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

$endgroup$

share|cite|improve this question

edited Mar 25 at 14:53

NumThcurious

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

share|cite|improve this question

edited Mar 25 at 14:53

NumThcurious

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013

asked Mar 24 at 16:29

NumThcuriousNumThcurious

1013