2022 China's Math GaoKao(College Entrance Exams): Problem 16 (eclipse, conic section, Vieta formula)

From 2022 China's Math GaoKao (College Entrance Exams): Problem 16

Topics:eclipse, conic section, Vieta formula

[kv-szgn]

It would be more easier if you consider a line y=mx+c
Its slope could be find by that perpendicular realtion, it passes through one of the focus(-a/2, 0) and now if we solve eq y=mx+c and ellipse we will get roots as the points on ellipse (but dont solve it by now), now bu distance formula Root((x1-x2)*2+(y1-y2)*2)
We can find x1-x2 (by difference of roots) and same for y
Ans we could get our result

[UzziWallendorf]

1/³a² b²-> ¾->¼->⅓¾->3/3/6/3/3/3->3->13