Ellipse tangent line using projections

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SUMMARY

The discussion focuses on proving that a line drawn from the perpendicular projections of intersection points on the major and minor axes of an ellipse is the tangent line at a given point R. The participants discuss the geometric properties of the ellipse, specifically the relationships between points P, P', Q, and Q' and their projections R', R'', and the intersection points of lines PR and QR. The confusion arises from the misunderstanding of the intersection of line segments P'R' and PR, as well as Q'R'' and QR.

PREREQUISITES
  • Understanding of ellipse geometry and properties
  • Knowledge of perpendicular projections in coordinate geometry
  • Familiarity with intersection points of lines and segments
  • Basic skills in sketching geometric figures for visualization
NEXT STEPS
  • Study the properties of ellipses, focusing on major and minor axes
  • Learn about geometric projections and their applications in coordinate geometry
  • Explore the concept of tangent lines in relation to conic sections
  • Practice sketching and analyzing geometric figures to enhance visualization skills
USEFUL FOR

Students studying geometry, particularly those interested in conic sections, as well as educators and anyone looking to deepen their understanding of ellipse properties and tangent lines.

arthur werbrouck
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Hi :)

The question is in dutch so i'l translate it.

on an ellipse E with vertex P and P' on the major axis and vertex Q and Q' on the minor axis. chose R(x1,y1), the projection of R on the major axis is R' and on the minor axis is R''. Define the perpendicular projection of the intrersection point of PR and P'R' on the major axis. And define the perpendicular projection of the intersection point of QR and Q'R'' on the minor axis. prove that the line drawn from these two projections is the tangent line of R.

I Get stuck every time. Sorry if the awnser is obvious, I'm only 16. I attached a quick sketch i made and sorry if there are translation errors.kind regards Arthur

this is the sketch
 
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arthur werbrouck said:
Define the perpendicular projection of the intrersection point of PR and P'R' on the major axis.
Something must have got lost in translation. The line segment P'R' lies on the major axis and does not intersect the line segment PR. If we extend P'R' to intersect PR then it does so at P, which generates no new points.

Similarly, Q'R'' lies on the minor axis and does not intersect QR'.

I suggest posting a diagram.
 

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