A Perimeter That Knows the Major Axis
Difficulty: Medium · Topics: Geometry, Conic Sections, Ellipse
Katta o‘qni biladigan perimetr
A line through the point F(3, 0) meets the ellipse dfracx^2a^2+dfracy^2b^2=1, whose foci are F(3, 0) and F'(-3, 0), at the two points A and B.
graph "x":[-6.4,6.4],"y":[-5.2,5.2],"xTicks":[],"yTicks":[],"label":"An ellipse with foci F and F′, and a chord AB through F forming triangle AF′B","items":["type":"ellipse","at":[0,0],"rx":5,"ry":4,"type":"polygon","points":[[2.5,3.464],[-3,0],[3.421,-2.917]],"ink":true,"fill":0.1,"type":"point","at":[2.5,3.464],"label":"A","anchor":"ne","type":"point","at":[3.421,-2.917],"label":"B","anchor":"se","type":"point","at":[3,0],"label":"F","anchor":"ne","ink":true,"type":"point","at":[-3,0],"label":"F′","anchor":"sw","ink":true]
If the perimeter of triangle AF'B is 20, find the value of ab. (Here a b 0.)
F(3, 0) nuqtadan o‘tuvchi to‘g‘ri chiziq fokuslari F(3, 0) va F'(-3, 0) bo‘lgan dfracx^2a^2+dfracy^2b^2=1 ellipsni ikki A va B nuqtada kesadi.
graph "x":[-6.4,6.4],"y":[-5.2,5.2],"xTicks":[],"yTicks":[],"label":"An ellipse with foci F and F′, and a chord AB through F forming triangle AF′B","items":["type":"ellipse","at":[0,0],"rx":5,"ry":4,"type":"polygon","points":[[2.5,3.464],[-3,0],[3.421,-2.917]],"ink":true,"fill":0.1,"type":"point","at":[2.5,3.464],"label":"A","anchor":"ne","type":"point","at":[3.421,-2.917],"label":"B","anchor":"se","type":"point","at":[3,0],"label":"F","anchor":"ne","ink":true,"type":"point","at":[-3,0],"label":"F′","anchor":"sw","ink":true]
Agar AF'B uchburchakning perimetri 20 bo‘lsa, ab ning qiymatini toping. (Bu yerda a b 0.)