Infinite line intersecting parabola only once

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SUMMARY

An infinite continuous line represented by the equation ax + by = c will only intersect a parabola once if it is a vertical line. This conclusion is based on the observation that any line with a non-zero slope will eventually cross the parabola again due to the nature of its curvature. The discussion also highlights that while tangent lines touch the parabola at a single point, they do not meet the criteria of crossing it. Therefore, the only valid solution for a single intersection is a vertical line parallel to the axis of the parabola.

PREREQUISITES
  • Understanding of parabola properties and equations
  • Knowledge of linear equations in the form ax + by = c
  • Familiarity with the concept of tangent lines
  • Basic principles of calculus related to slopes and intersections
NEXT STEPS
  • Study the properties of parabolas and their equations
  • Learn about the conditions for tangency between lines and curves
  • Explore the implications of vertical lines in coordinate geometry
  • Investigate the concept of limits and slopes in calculus
USEFUL FOR

Mathematics students, educators, and anyone interested in the geometric properties of curves and lines, particularly in the context of calculus and algebra.

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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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