High School Ellipse tangent line using projections

[arthur werbrouck]

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

 

[andrewkirk]

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.