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

In summary: 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!
  • #1
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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
View attachment 5721
View attachment 5722
https://www.physicsforums.com/attachments/5723
https://www.physicsforums.com/attachments/5724
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  • #2
Hi Peter,

Peter said:
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!
 
  • #3
GJA said:
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
 
  • #4
Hi Peter,

Peter said:
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!
 

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

1. What is the purpose of Exercise 3(a) in Section 8.1 of Cox et al?

Exercise 3(a) in Section 8.1 of Cox et al is designed to help students understand and apply the concepts of regression and correlation in a real-world scenario. It involves analyzing data and using statistical methods to draw conclusions about the relationship between variables.

2. What background knowledge is required to complete Exercise 3(a)?

To successfully complete Exercise 3(a), students should have a solid understanding of basic statistical concepts such as mean, median, and standard deviation. They should also have a working knowledge of regression and correlation analysis.

3. What skills will students gain from completing Exercise 3(a)?

By completing Exercise 3(a), students will develop skills in data analysis, interpreting statistical results, and drawing conclusions based on data. They will also gain a deeper understanding of regression and correlation and how they can be applied in real-world scenarios.

4. How can students approach Exercise 3(a) effectively?

To approach Exercise 3(a) effectively, students should carefully read the instructions and familiarize themselves with the data provided. They should also review the relevant concepts and formulas before beginning the exercise. It may also be helpful to work in a group or seek assistance from a tutor or instructor.

5. What are some potential challenges students may face when completing Exercise 3(a)?

Students may face challenges when completing Exercise 3(a) if they have difficulty understanding the statistical concepts or interpreting the results. They may also struggle with the data analysis process or encounter technical difficulties with the software or tools used. Seeking help from a tutor or instructor can help overcome these challenges.

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