Infinite line intersecting parabola only once

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Homework Help Overview

The problem involves determining the conditions under which an infinite continuous line, represented by the equation ax + by = c, intersects a parabola only once. The discussion centers on the nature of the line and its relationship to the parabola.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants explore the idea of vertical lines as potential solutions and question whether other types of lines could also intersect the parabola only once. There is a discussion about the distinction between lines that merely touch the parabola and those that cross it.

Discussion Status

The discussion is ongoing, with participants sharing their thoughts and questioning the assumptions made about the nature of the intersections. Some guidance has been offered regarding the characteristics of lines that intersect the parabola, but no consensus has been reached.

Contextual Notes

There is a mention of previous assignments involving tangent lines, which may influence the participants' understanding of the current problem. The original poster expresses uncertainty about their conclusions and seeks validation.

FiveAlive
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I think I know the answer to this problem however I am not 100% sure.

I must describe the conditions in which an infinite continuous line (ax+by=c) will only cross a parabola once. I believe the only answer is a straight, vertical line because while the slope of the parabola may approach zero and become extremely small it will never be zero. Therefore if the line has any slope at all, not matter how small, it will eventually cross the parabola again.

Am I correct in this conclusion? Are there options besides a vertical line?

Thanking you in advance,
Linus
 
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Should I assume since my post has many views but no replies that I am on the mark?
 
Well what about a line that just touches the parabola at one point?
 
I think the line must cross the parabola, not just share a point like a tangent line. I only assume this because we did tangent lines a while ago and it seems strange to have an assignment solely about finding them.
 
It is true that a line that crosses a parabola and is not tangent and which crosses it only once must be parallel to the axis of the parabola. But your proof of it is not very convincing. If there's no need to really prove it, then that's probably fine.
 

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