How Do I Find P1 and P2 for a Parametrized Curve?

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SUMMARY

The discussion focuses on determining points P1 and P2 for a parametrized curve defined by the equation ax² + bxy + cy² + dx + ex + f = 0. The user seeks to express coefficients a and b in terms of variables x, L, and λ, utilizing trigonometric functions of angle θ. The conversation emphasizes the importance of not assuming the curve is a conic section without further analysis. The user plans to continue exploring this problem after class.

PREREQUISITES
  • Understanding of parametrized curves and their equations.
  • Familiarity with conic sections and their properties.
  • Knowledge of trigonometric functions and their applications in geometry.
  • Basic algebraic manipulation skills to derive relationships between variables.
NEXT STEPS
  • Study the derivation of coefficients in conic sections using parametric equations.
  • Learn about the application of trigonometric functions in geometric problems.
  • Explore methods for analyzing curves without assuming their type.
  • Investigate the use of software tools for visualizing parametrized curves.
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Mathematicians, physics students, and anyone involved in geometric modeling or curve analysis will benefit from this discussion.

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Question: Attached.

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I know I have to find P1 and p2 depending on x and L to satisfy

ax^2 + bxy + cy^2 + dx + ex + f =0

I have tried looking at the ladder when resting on axis to get a better picture but cannot see where to go next.
 

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You don't want to start with the general equation of a conic because you don't know ahead of time it even is a conic. I know, the question tells you that, but you shouldn't assume it.

Here's a picture of the slanted ladder:

ladder.jpg


Figure out a and b in terms of x, L, and λ. You will need to use the trig functions of θ which you can also get in terms of x, L, and λ. That should get you started.
 
Thanks so much! I am going to work more on this today after class.
 

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