Is there a parabola which is similar to a branch of hyperbola?

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Parabola and a branch of hyperbola, visually looks similar.



The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an arbitrary line the parabola will rush past that line and keep going away from it.



But still, is there any parabola which is exactly what a branch of hyperbola looks like?










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    Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
    – bof
    Dec 6 at 5:28














up vote
4
down vote

favorite












Parabola and a branch of hyperbola, visually looks similar.



The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an arbitrary line the parabola will rush past that line and keep going away from it.



But still, is there any parabola which is exactly what a branch of hyperbola looks like?










share|cite|improve this question



















  • 3




    Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
    – bof
    Dec 6 at 5:28












up vote
4
down vote

favorite









up vote
4
down vote

favorite











Parabola and a branch of hyperbola, visually looks similar.



The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an arbitrary line the parabola will rush past that line and keep going away from it.



But still, is there any parabola which is exactly what a branch of hyperbola looks like?










share|cite|improve this question















Parabola and a branch of hyperbola, visually looks similar.



The only difference I find is that, when x tends to infinity, hyperbola approaches a straight line (asymptote). Whereas if I draw an arbitrary line the parabola will rush past that line and keep going away from it.



But still, is there any parabola which is exactly what a branch of hyperbola looks like?







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edited Dec 6 at 11:32

























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  • 3




    Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
    – bof
    Dec 6 at 5:28












  • 3




    Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
    – bof
    Dec 6 at 5:28







3




3




Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
– bof
Dec 6 at 5:28




Since all parabolas have the same shape, one parabola isn't going to look more hyperbolic than anothesr.
– bof
Dec 6 at 5:28










3 Answers
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4
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The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.



Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.



Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.



So no, you cannot make a parabola look like a branch of a typical hyperbola.
But you can make a branch of a hyperbola look almost like a parabola.



The match will still not be exact. You might as well ask for a positive number that is exactly zero.






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    No parabola has an asymptote, while every branch of a hyperbola has two asymptotes. Therefore, there can never be a parabola that looks exactly like a branch of a hyperbola.






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      Actually parabola is a special case of hyperbola, where eccentricity tends to 1.



      Consider this equation of parabola. $$(x-2)^2+y^2=1•dfrac1$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.



      enter image description here



      Guess what this is



      enter image description here



      Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image.



      enter image description here



      How far the second focus has reached, near to $-1times 10^7$. As the eccentricity tends to 1, the second focus tends to $-infty$ and the hyperbola looks more like a parabola.






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        3 Answers
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        3 Answers
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        The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.



        Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.



        Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.



        So no, you cannot make a parabola look like a branch of a typical hyperbola.
        But you can make a branch of a hyperbola look almost like a parabola.



        The match will still not be exact. You might as well ask for a positive number that is exactly zero.






        share|cite|improve this answer
























          up vote
          4
          down vote













          The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.



          Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.



          Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.



          So no, you cannot make a parabola look like a branch of a typical hyperbola.
          But you can make a branch of a hyperbola look almost like a parabola.



          The match will still not be exact. You might as well ask for a positive number that is exactly zero.






          share|cite|improve this answer






















            up vote
            4
            down vote










            up vote
            4
            down vote









            The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.



            Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.



            Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.



            So no, you cannot make a parabola look like a branch of a typical hyperbola.
            But you can make a branch of a hyperbola look almost like a parabola.



            The match will still not be exact. You might as well ask for a positive number that is exactly zero.






            share|cite|improve this answer












            The only differences between two parabolas are location, orientation, and scaling factor. As noted in a comment, they all have the same shape.



            Hyperbolas, however, come in many different shapes. Some are asymptotic to a pair of perpendicular lines. Others live inside a much larger or much smaller angle between their asymptotic lines.



            Now consider a sequence of hyperbolas constructed as follows. We put one vertex of the hyperbola at a fixed point and move the other vertex away, allowing the angle between the asymptotic lines to approach zero as the other vertex goes off to infinity. If we cleverly balance the rates at which the angle gets smaller and the other vertex gets farther, the hyperbolas will approach the shape of a parabola.



            So no, you cannot make a parabola look like a branch of a typical hyperbola.
            But you can make a branch of a hyperbola look almost like a parabola.



            The match will still not be exact. You might as well ask for a positive number that is exactly zero.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 6 at 5:38









            David K

            52k340115




            52k340115




















                up vote
                3
                down vote













                No parabola has an asymptote, while every branch of a hyperbola has two asymptotes. Therefore, there can never be a parabola that looks exactly like a branch of a hyperbola.






                share|cite|improve this answer
























                  up vote
                  3
                  down vote













                  No parabola has an asymptote, while every branch of a hyperbola has two asymptotes. Therefore, there can never be a parabola that looks exactly like a branch of a hyperbola.






                  share|cite|improve this answer






















                    up vote
                    3
                    down vote










                    up vote
                    3
                    down vote









                    No parabola has an asymptote, while every branch of a hyperbola has two asymptotes. Therefore, there can never be a parabola that looks exactly like a branch of a hyperbola.






                    share|cite|improve this answer












                    No parabola has an asymptote, while every branch of a hyperbola has two asymptotes. Therefore, there can never be a parabola that looks exactly like a branch of a hyperbola.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 6 at 5:35









                    Robert Howard

                    1,9181822




                    1,9181822




















                        up vote
                        2
                        down vote













                        Actually parabola is a special case of hyperbola, where eccentricity tends to 1.



                        Consider this equation of parabola. $$(x-2)^2+y^2=1•dfrac1$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.



                        enter image description here



                        Guess what this is



                        enter image description here



                        Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image.



                        enter image description here



                        How far the second focus has reached, near to $-1times 10^7$. As the eccentricity tends to 1, the second focus tends to $-infty$ and the hyperbola looks more like a parabola.






                        share|cite|improve this answer


























                          up vote
                          2
                          down vote













                          Actually parabola is a special case of hyperbola, where eccentricity tends to 1.



                          Consider this equation of parabola. $$(x-2)^2+y^2=1•dfrac1$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.



                          enter image description here



                          Guess what this is



                          enter image description here



                          Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image.



                          enter image description here



                          How far the second focus has reached, near to $-1times 10^7$. As the eccentricity tends to 1, the second focus tends to $-infty$ and the hyperbola looks more like a parabola.






                          share|cite|improve this answer
























                            up vote
                            2
                            down vote










                            up vote
                            2
                            down vote









                            Actually parabola is a special case of hyperbola, where eccentricity tends to 1.



                            Consider this equation of parabola. $$(x-2)^2+y^2=1•dfrac1$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.



                            enter image description here



                            Guess what this is



                            enter image description here



                            Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image.



                            enter image description here



                            How far the second focus has reached, near to $-1times 10^7$. As the eccentricity tends to 1, the second focus tends to $-infty$ and the hyperbola looks more like a parabola.






                            share|cite|improve this answer














                            Actually parabola is a special case of hyperbola, where eccentricity tends to 1.



                            Consider this equation of parabola. $$(x-2)^2+y^2=1•dfrac1$$ This is a parabola with focus at $(2,0)$ and equation of directrix $x=0$. This looks something like this.



                            enter image description here



                            Guess what this is



                            enter image description here



                            Looks similar to the above parabola, but it isn't. It's a hyperbola with same focus, same equation of directrix but with eccentricity 1.000001. Don't believe me see this zoomed out image.



                            enter image description here



                            How far the second focus has reached, near to $-1times 10^7$. As the eccentricity tends to 1, the second focus tends to $-infty$ and the hyperbola looks more like a parabola.







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited Dec 6 at 11:33

























                            answered Dec 6 at 10:08









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