Finding Tangent Lines for Parabolas: A Mathematical Approach

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Discussion Overview

The discussion revolves around finding two tangent lines to a parabola defined by the equation resulting from the product of two linear functions. Participants explore the mathematical approach to determining these lines, including the conditions necessary for tangency and the implications of slopes.

Discussion Character

  • Mathematical reasoning
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant suggests that the two lines must have opposite slopes or at least one positive and one negative slope to ensure they are tangent to the parabola.
  • Another participant explains that for a parabola expressed as the product of two lines, it must touch each line at the zeros of the parabola, and differentiating the product can help find conditions for tangency.
  • There is mention of a trivial case where both lines are at y=0, which is dismissed as invalid for the context of the problem.
  • A participant notes that the two lines must intersect at the same y-coordinate of 0.5.
  • One participant expresses uncertainty about how to proceed without a formula for tangency, indicating a struggle with differentiation.
  • Another participant asserts that calculus is not necessary to determine tangency, suggesting that the number of intersection points can suffice.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of calculus for solving the problem, with some asserting it is essential while others believe it can be approached without it. The discussion remains unresolved regarding the best method to find the tangent lines.

Contextual Notes

There is uncertainty regarding the foundational knowledge required for the problem, particularly in differentiation and the conditions for tangency. Participants also mention the potential complexity of the topic given the participant's educational level.

Who May Find This Useful

This discussion may be useful for students in algebra or early high school mathematics, particularly those exploring the relationships between linear functions and parabolas.

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Ok, I need to find 2 lines that when multiplied (y=x*y=x yields y=x^2)
create a parabola in which those two lines are tangent to it. My problem is that I have no idea where to start. I think that the two lines have to have opposite slopes, or at least one slope must be positive and one must be negative, else both lines would follow the same direction, and there is an obvious problem there...
 
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This was surprisingly tricky. The first thing to realize is that if a parabola can be expressed as the product of two lines ax+b and cx+d, then the parabola's equation is (ax+b)(cx+d) and the lines factor it. The next thing to realize is that the parabola must touch a line at each zero of the parabola, and if the line is not tangent at the zero it is not tangent anywhere. Now differentiate (ax+b)(cx+d) (it's easier later on if you do it using the product rule rather than expanding it out first). Then, you must find a simple condition to make the slope of the parabola at the zeroes appropriate, and then solve that condition.

There is also the trivial case of y1 = 0 = y2 but since you ask for a parabola I don't think that is a valid answer.

Edit: it's interesting, the two lines must always intersect at the same y coordinate, 0.5.

What class is this for?
 
Last edited:
It's for my Honors Algebra II class. Not everyone has to do it, my teacher and I are working together on upper level math like this. I can generally get the stuff we discuss, but some of it goes right over my head, as I'm only in 9th grade. But thanks a lot for the help on this one, it's greatly appriceated.
 
Well, if you're only in algebra II I don't see how you can do it without differentiating. Are you given a formula to tell if a line is tangent to a parabola?
 
Nope, and that's where I'm stuck at the moment. I see the (ax+b)(cx+d)=y, but I've got nothing after that... I tried expanding it out which gives
y=acx^2+adx+bcx+bd and that yields nothing that I can see.
 
If you don't know how to take a derivative and you have not been given a formula to tell when a line is tangent to a parabola, you can't do the problem.
 
Ok, thanks anyway then. I guess I'll look for that formula, as I took a look at derivatives, and I doubt I could learn that without having a foundation in Calculus let alone Trig or Euclidian Geometry.
 
You don't need calculus to find out when a line is tangent to a parabola. You just need to know how many times they intersect.
 
Oh, right! It is also much simpler to do it from that perspective.
 

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