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Say we wanted to check the behaviour of the following hyperbola when $x,y$ are very large $$\frac{x^2}{a^2}=\frac{y^2}{b^2}+1$$ Clearly 1 becomes insignificant when compared to $x,y$ so we get$$y=\pm\frac{b}{a}x$$ So we can say that when $x,y$ are very large our hyperbola approximates to a line but when when we try to solve this line with our hyperbola we get no solution so we can claim that this is its asymptote(because it approximately behaves like our curve when $x,y$ are very large and also it never intersects our curve) (One may also check another similar way given as an answer to this question Discriminant of a Conic Section) What I wanted to know is that when we try to do something similar with a parabola having equation as$$y^2=4ax$$ We get $$y=0$$

Why do we get such things? Does it even has any meaning ? What does this tell us about the curve?

Anshu Gupta
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    This is not quite the same thing, as $x\to\infty$ as well. What is really going on is clearest if we look in projective space, including the line at infinity. What you see is that the parabola does indeed intersect the line at infinity at $y=0$, because the slope of the parabola approaches $0$ as you go off to infinity. Officially, you write $x^2/a^2=y^2/b^2+z^2$ and $y^2=4axz$, making the equations homogeneous of degree $2$, and then look at infinity by setting $z=0$. – Ted Shifrin Dec 19 '23 at 20:20
  • I agree that the slope of the parabola goes to 0,but I'm not able to understand other things which you said may you please elaborate your point sir @TedShifrin – Anshu Gupta Dec 20 '23 at 07:00

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