Peter Needs Help on Cox et al - Section 8.1, Exercise 3(a)

[Math Amateur]

I am reading the undergraduate introduction to algebraic geometry entitled "Ideals, Varieties and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra (Third Edition) by David Cox, John Little and Donal O'Shea ... ...

I am currently focused on Chapter 8, Section 1: The Projective Plane ... ... and need help getting started with Exercise 3(a) ... Exercise 3 in Section 8.1 reads as follows:View attachment 5719I would very much appreciate someone helping me to start Exercise 3(a) ... ...

Peter======================================================================To give readers of the above post some idea of the context of the exercise and also the notation I am providing some relevant text from Cox et al ... ... as follows:https://www.physicsforums.com/attachments/5720
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https://www.physicsforums.com/attachments/5723
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[GJA]

Hi Peter,

I am currently focused on Chapter 8, Section 1: The Projective Plane ... ... and need help getting started with Exercise 3(a) ...

If it helps at all, it may be of value to note that knowledge of the projective plane is not needed to solve Exercise 3(a). One way to solve the problem is to write the equation of the line in $\mathbb{R}^{2}$ with slope $t$. The line and the hyperbola will intersect provided that there is an ordered pair/point $(x,y)$ that satisfies both the line and hyperbola equations simultaneously. So, given $t$, you want to check that the point $(x,y) = ((1+t^2)/(1-t^2), 2t/(1-t^2))$ that sits on the hyperbola satisfies the equation for the line you wrote down. If it does, then the line and hyperbola DO intersect at that point $(x,y)$.

Let me know how it goes. Good luck!

 

[Math Amateur]

Hi Peter,
If it helps at all, it may be of value to note that knowledge of the projective plane is not needed to solve Exercise 3(a). One way to solve the problem is to write the equation of the line in $\mathbb{R}^{2}$ with slope $t$. The line and the hyperbola will intersect provided that there is an ordered pair/point $(x,y)$ that satisfies both the line and hyperbola equations simultaneously. So, given $t$, you want to check that the point $(x,y) = ((1+t^2)/(1-t^2), 2t/(1-t^2))$ that sits on the hyperbola satisfies the equation for the line you wrote down. If it does, then the line and hyperbola DO intersect at that point $(x,y)$.

Let me know how it goes. Good luck!

Thanks GJA ... yes, was very straightforward, as you say ...A line of slope \(\displaystyle t\) going through the point \(\displaystyle (-1,0)\) is \(\displaystyle y = tx + t\) ... ( or \(\displaystyle t = \frac{y}{(x+1)} )\) ...... and the point \(\displaystyle x = \frac{1+t^2}{1-t^2} , \ y = \frac{2t}{1 - t^2}\) lies on the line ... so the hyperbola and line intersect at \(\displaystyle (x,y)\) ... ...
BUT ... how does that enable us to answer part (b) ... ...... that is ... how do we use part (a) to explain why \(\displaystyle t = \pm 1\) maps to the points at \infty corresponding to the asymptotes of the hyperbola ... ... Can you help ...

Peter

 

[GJA]

Hi Peter,

BUT ... how does that enable us to answer part (b) ... ...... that is ... how do we use part (a) to explain why \(\displaystyle t = \pm 1\) maps to the points at \infty corresponding to the asymptotes of the hyperbola ... ...

I really think that drawing a picture of the line and the hyperbola on the same coordinate axis might provide insight into why this is the case. As $t$ approaches $\pm 1$ try to note the behaviour of the hyperbola relative to the line.

Good luck!