A Midpoint Equidistant from Three Points
Difficulty: Medium · Topics: Geometry, Conic Sections, Ellipse
Uch nuqtadan teng uzoqlikdagi o‘rta nuqta
The ellipse dfracx^2a^2+dfracy^2b^2=1 (a b 0) has foci F(c, 0) and F'(-c, 0), where c 0. A line through F' meets the ellipse at the two points P and Q.
graph "x":[-7.4,7.4],"y":[-5.2,5.2],"xTicks":[],"yTicks":[],"label":"An ellipse with a chord PQ through the focus F′, and the midpoint M of FQ","items":["type":"ellipse","at":[0,0],"rx":6,"ry":4,"type":"line","from":[-2.683,3.578],"to":[-5.367,-1.789],"ink":true,"type":"line","from":[4.472,0],"to":[-5.367,-1.789],"ink":true,"type":"line","from":[4.472,0],"to":[-2.683,3.578],"ink":true,"type":"line","from":[-2.683,3.578],"to":[-0.447,-0.894],"ink":true,"dash":true,"type":"point","at":[-2.683,3.578],"label":"P","anchor":"n","type":"point","at":[-5.367,-1.789],"label":"Q","anchor":"sw","type":"point","at":[4.472,0],"label":"F","anchor":"se","ink":true,"type":"point","at":[-4.472,0],"label":"F′","anchor":"ne","ink":true,"type":"point","at":[-0.447,-0.894],"label":"M","anchor":"se","ink":true]
If overlinePQ=6 and the midpoint M of the segment FQ satisfies overlineFM=overlinePM=5, find the length of the minor axis of this ellipse.
dfracx^2a^2+dfracy^2b^2=1 (a b 0) ellipsning fokuslari F(c, 0) va F'(-c, 0), bunda c 0. F' dan o‘tuvchi to‘g‘ri chiziq ellipsni ikki P va Q nuqtada kesadi.
graph "x":[-7.4,7.4],"y":[-5.2,5.2],"xTicks":[],"yTicks":[],"label":"An ellipse with a chord PQ through the focus F′, and the midpoint M of FQ","items":["type":"ellipse","at":[0,0],"rx":6,"ry":4,"type":"line","from":[-2.683,3.578],"to":[-5.367,-1.789],"ink":true,"type":"line","from":[4.472,0],"to":[-5.367,-1.789],"ink":true,"type":"line","from":[4.472,0],"to":[-2.683,3.578],"ink":true,"type":"line","from":[-2.683,3.578],"to":[-0.447,-0.894],"ink":true,"dash":true,"type":"point","at":[-2.683,3.578],"label":"P","anchor":"n","type":"point","at":[-5.367,-1.789],"label":"Q","anchor":"sw","type":"point","at":[4.472,0],"label":"F","anchor":"se","ink":true,"type":"point","at":[-4.472,0],"label":"F′","anchor":"ne","ink":true,"type":"point","at":[-0.447,-0.894],"label":"M","anchor":"se","ink":true]
Agar overlinePQ=6 va FQ kesmaning M o‘rta nuqtasi overlineFM=overlinePM=5 ni qanoatlantirsa, bu ellipsning kichik o‘qi uzunligini toping.