How many parabolas can pass through two given points?












2












$begingroup$


I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.



My question is, how many?






geometry







share|cite|improve this question









$endgroup$








  • 14




    $begingroup$
    Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
    $endgroup$
    – Blue
    yesterday








  • 1




    $begingroup$
    In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
    $endgroup$
    – Oscar Bravo
    yesterday


















2












$begingroup$


I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.



My question is, how many?






geometry







share|cite|improve this question









$endgroup$








  • 14




    $begingroup$
    Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
    $endgroup$
    – Blue
    yesterday








  • 1




    $begingroup$
    In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
    $endgroup$
    – Oscar Bravo
    yesterday
















2












2








2


1



$begingroup$


I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.



My question is, how many?






geometry







share|cite|improve this question









$endgroup$




I know that in order to uniquely determine a parabola, we require 3 points. So naturally, 2 points will have multiple possible parabolas pass through them.



My question is, how many?






geometry




geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









Dhruv GuptaDhruv Gupta

696




696








  • 14




    $begingroup$
    Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
    $endgroup$
    – Blue
    yesterday








  • 1




    $begingroup$
    In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
    $endgroup$
    – Oscar Bravo
    yesterday
















  • 14




    $begingroup$
    Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
    $endgroup$
    – Blue
    yesterday








  • 1




    $begingroup$
    In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
    $endgroup$
    – Oscar Bravo
    yesterday










14




14




$begingroup$
Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
$endgroup$
– Blue
yesterday






$begingroup$
Correction: In order to uniquely determine a parabola with an axis in a particular direction, we require 3 points. For why three points aren't sufficient in general, for instance, see, for instance, this answer.
$endgroup$
– Blue
yesterday






1




1




$begingroup$
In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
$endgroup$
– Oscar Bravo
yesterday






$begingroup$
In questions like this, the possible answers are, zero, one and infinity. You already said it's more than one, so it must be infinity.
$endgroup$
– Oscar Bravo
yesterday












3 Answers
3






active

oldest

votes


















6












$begingroup$

Infinitely many.
Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Thank you! When you put it that way, it seems so obvious. Thanks again.
    $endgroup$
    – Dhruv Gupta
    yesterday






  • 23




    $begingroup$
    You don't get a different parabola for every point. Still infinity, though.
    $endgroup$
    – Spitemaster
    yesterday










  • $begingroup$
    You can set a third point anywhere not on any parabola you've yet considered.
    $endgroup$
    – Cai
    yesterday






  • 1




    $begingroup$
    Yeah, naturally, you are right. I rewrote my answer.
    $endgroup$
    – Mundron Schmidt
    yesterday



















9












$begingroup$

Any size of parabola, with infinitely many orientations for each parabola!



Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.



Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)



Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,infty)$ and an angle in the range $(0,π)cup(-π, 0)$.



Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    +1 very pretty geometry. Much nicer than than the algebraic analysis.
    $endgroup$
    – Ethan Bolker
    yesterday



















1












$begingroup$

General equation of a conic section:



$small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;



where $A,B,C,D,E,F$ are constants.



Necessary condition for a parabola :



$B^2-4AC =0$, which leaves $5$ constants to be determined.



Cf. Blue's link in his comment above.






share|cite|improve this answer











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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    Infinitely many.
    Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Thank you! When you put it that way, it seems so obvious. Thanks again.
      $endgroup$
      – Dhruv Gupta
      yesterday






    • 23




      $begingroup$
      You don't get a different parabola for every point. Still infinity, though.
      $endgroup$
      – Spitemaster
      yesterday










    • $begingroup$
      You can set a third point anywhere not on any parabola you've yet considered.
      $endgroup$
      – Cai
      yesterday






    • 1




      $begingroup$
      Yeah, naturally, you are right. I rewrote my answer.
      $endgroup$
      – Mundron Schmidt
      yesterday
















    6












    $begingroup$

    Infinitely many.
    Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Thank you! When you put it that way, it seems so obvious. Thanks again.
      $endgroup$
      – Dhruv Gupta
      yesterday






    • 23




      $begingroup$
      You don't get a different parabola for every point. Still infinity, though.
      $endgroup$
      – Spitemaster
      yesterday










    • $begingroup$
      You can set a third point anywhere not on any parabola you've yet considered.
      $endgroup$
      – Cai
      yesterday






    • 1




      $begingroup$
      Yeah, naturally, you are right. I rewrote my answer.
      $endgroup$
      – Mundron Schmidt
      yesterday














    6












    6








    6





    $begingroup$

    Infinitely many.
    Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.






    share|cite|improve this answer











    $endgroup$



    Infinitely many.
    Draw a line between your two points. Now chose any points on the line as your third point. Since two different points on the line will give you different parabola, you get infinitely many different parabola.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited yesterday

























    answered yesterday









    Mundron SchmidtMundron Schmidt

    7,3942729




    7,3942729








    • 1




      $begingroup$
      Thank you! When you put it that way, it seems so obvious. Thanks again.
      $endgroup$
      – Dhruv Gupta
      yesterday






    • 23




      $begingroup$
      You don't get a different parabola for every point. Still infinity, though.
      $endgroup$
      – Spitemaster
      yesterday










    • $begingroup$
      You can set a third point anywhere not on any parabola you've yet considered.
      $endgroup$
      – Cai
      yesterday






    • 1




      $begingroup$
      Yeah, naturally, you are right. I rewrote my answer.
      $endgroup$
      – Mundron Schmidt
      yesterday














    • 1




      $begingroup$
      Thank you! When you put it that way, it seems so obvious. Thanks again.
      $endgroup$
      – Dhruv Gupta
      yesterday






    • 23




      $begingroup$
      You don't get a different parabola for every point. Still infinity, though.
      $endgroup$
      – Spitemaster
      yesterday










    • $begingroup$
      You can set a third point anywhere not on any parabola you've yet considered.
      $endgroup$
      – Cai
      yesterday






    • 1




      $begingroup$
      Yeah, naturally, you are right. I rewrote my answer.
      $endgroup$
      – Mundron Schmidt
      yesterday








    1




    1




    $begingroup$
    Thank you! When you put it that way, it seems so obvious. Thanks again.
    $endgroup$
    – Dhruv Gupta
    yesterday




    $begingroup$
    Thank you! When you put it that way, it seems so obvious. Thanks again.
    $endgroup$
    – Dhruv Gupta
    yesterday




    23




    23




    $begingroup$
    You don't get a different parabola for every point. Still infinity, though.
    $endgroup$
    – Spitemaster
    yesterday




    $begingroup$
    You don't get a different parabola for every point. Still infinity, though.
    $endgroup$
    – Spitemaster
    yesterday












    $begingroup$
    You can set a third point anywhere not on any parabola you've yet considered.
    $endgroup$
    – Cai
    yesterday




    $begingroup$
    You can set a third point anywhere not on any parabola you've yet considered.
    $endgroup$
    – Cai
    yesterday




    1




    1




    $begingroup$
    Yeah, naturally, you are right. I rewrote my answer.
    $endgroup$
    – Mundron Schmidt
    yesterday




    $begingroup$
    Yeah, naturally, you are right. I rewrote my answer.
    $endgroup$
    – Mundron Schmidt
    yesterday











    9












    $begingroup$

    Any size of parabola, with infinitely many orientations for each parabola!



    Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.



    Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)



    Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,infty)$ and an angle in the range $(0,π)cup(-π, 0)$.



    Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      +1 very pretty geometry. Much nicer than than the algebraic analysis.
      $endgroup$
      – Ethan Bolker
      yesterday
















    9












    $begingroup$

    Any size of parabola, with infinitely many orientations for each parabola!



    Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.



    Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)



    Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,infty)$ and an angle in the range $(0,π)cup(-π, 0)$.



    Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      +1 very pretty geometry. Much nicer than than the algebraic analysis.
      $endgroup$
      – Ethan Bolker
      yesterday














    9












    9








    9





    $begingroup$

    Any size of parabola, with infinitely many orientations for each parabola!



    Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.



    Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)



    Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,infty)$ and an angle in the range $(0,π)cup(-π, 0)$.



    Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.






    share|cite|improve this answer











    $endgroup$



    Any size of parabola, with infinitely many orientations for each parabola!



    Suppose the points are distance $d$ apart. Call them $A$ and $B$. Take your chosen parabola and pick two points on it separated by distance $d$. Then place them on $A$ and $B$.



    Since you can slide the points along the parabola to wherever you like, the parabola can have any orientation that doesn't put its axis parallel to $AB$. (The sliding only gives you $180°$ worth of orientations, but you can reflect it in $AB$ to get the other $180°$.)



    Assuming points on the parabola have real numbers as their coordinates, its size and orientation can be described by two real numbers: a size in the range $(0,infty)$ and an angle in the range $(0,π)cup(-π, 0)$.



    Because of the way infinities work, this makes the set of parabolas through $A$ and $B$ uncountably infinite and the same size as the set of real nunbers.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited yesterday

























    answered yesterday









    timtfjtimtfj

    1,982419




    1,982419








    • 1




      $begingroup$
      +1 very pretty geometry. Much nicer than than the algebraic analysis.
      $endgroup$
      – Ethan Bolker
      yesterday














    • 1




      $begingroup$
      +1 very pretty geometry. Much nicer than than the algebraic analysis.
      $endgroup$
      – Ethan Bolker
      yesterday








    1




    1




    $begingroup$
    +1 very pretty geometry. Much nicer than than the algebraic analysis.
    $endgroup$
    – Ethan Bolker
    yesterday




    $begingroup$
    +1 very pretty geometry. Much nicer than than the algebraic analysis.
    $endgroup$
    – Ethan Bolker
    yesterday











    1












    $begingroup$

    General equation of a conic section:



    $small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;



    where $A,B,C,D,E,F$ are constants.



    Necessary condition for a parabola :



    $B^2-4AC =0$, which leaves $5$ constants to be determined.



    Cf. Blue's link in his comment above.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      General equation of a conic section:



      $small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;



      where $A,B,C,D,E,F$ are constants.



      Necessary condition for a parabola :



      $B^2-4AC =0$, which leaves $5$ constants to be determined.



      Cf. Blue's link in his comment above.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        General equation of a conic section:



        $small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;



        where $A,B,C,D,E,F$ are constants.



        Necessary condition for a parabola :



        $B^2-4AC =0$, which leaves $5$ constants to be determined.



        Cf. Blue's link in his comment above.






        share|cite|improve this answer











        $endgroup$



        General equation of a conic section:



        $small {Ax^2+Bxy +Cy^2 +Dx+Ey +F=0}$;



        where $A,B,C,D,E,F$ are constants.



        Necessary condition for a parabola :



        $B^2-4AC =0$, which leaves $5$ constants to be determined.



        Cf. Blue's link in his comment above.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited yesterday

























        answered yesterday









        Peter SzilasPeter Szilas

        11.1k2821




        11.1k2821






























            draft saved

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