如图,在x轴上有两点A(m,0),B(n,0)(n>m>0).分别过点A,点B作x轴的垂线,交抛物线y=x2于点C、点D.直线OC交直线BD于点E,直线OD交直线AC于点F,点E、点F的纵坐标分别记为yE,yF.特例探究填空
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![如图,在x轴上有两点A(m,0),B(n,0)(n>m>0).分别过点A,点B作x轴的垂线,交抛物线y=x2于点C、点D.直线OC交直线BD于点E,直线OD交直线AC于点F,点E、点F的纵坐标分别记为yE,yF.特例探究填空](/uploads/image/z/9934946-26-6.jpg?t=%E5%A6%82%E5%9B%BE%2C%E5%9C%A8x%E8%BD%B4%E4%B8%8A%E6%9C%89%E4%B8%A4%E7%82%B9A%EF%BC%88m%2C0%EF%BC%89%2CB%EF%BC%88n%2C0%EF%BC%89%EF%BC%88n%EF%BC%9Em%EF%BC%9E0%EF%BC%89%EF%BC%8E%E5%88%86%E5%88%AB%E8%BF%87%E7%82%B9A%2C%E7%82%B9B%E4%BD%9Cx%E8%BD%B4%E7%9A%84%E5%9E%82%E7%BA%BF%2C%E4%BA%A4%E6%8A%9B%E7%89%A9%E7%BA%BFy%3Dx2%E4%BA%8E%E7%82%B9C%E3%80%81%E7%82%B9D%EF%BC%8E%E7%9B%B4%E7%BA%BFOC%E4%BA%A4%E7%9B%B4%E7%BA%BFBD%E4%BA%8E%E7%82%B9E%2C%E7%9B%B4%E7%BA%BFOD%E4%BA%A4%E7%9B%B4%E7%BA%BFAC%E4%BA%8E%E7%82%B9F%2C%E7%82%B9E%E3%80%81%E7%82%B9F%E7%9A%84%E7%BA%B5%E5%9D%90%E6%A0%87%E5%88%86%E5%88%AB%E8%AE%B0%E4%B8%BAyE%2CyF%EF%BC%8E%E7%89%B9%E4%BE%8B%E6%8E%A2%E7%A9%B6%E5%A1%AB%E7%A9%BA)
如图,在x轴上有两点A(m,0),B(n,0)(n>m>0).分别过点A,点B作x轴的垂线,交抛物线y=x2于点C、点D.直线OC交直线BD于点E,直线OD交直线AC于点F,点E、点F的纵坐标分别记为yE,yF.特例探究填空
如图,在x轴上有两点A(m,0),B(n,0)(n>m>0).分别过点A,点B作x轴的垂线,交抛物线y=x2于点C、点D.直线OC交直线BD于点E,直线OD交直线AC于点F,点E、点F的纵坐标分别记为yE,yF.
特例探究
填空:
当m=1,n=2时,yE=,yF=;
当m=3,n=5时,yE=,yF=.
归纳证明
对任意m,n(n>m>0),猜想yE与yF的大小关系,并证明你的猜想.
拓展应用
(1)若将“抛物线y=x2”改为“抛物线y=ax2(a>0)”,其他条件不变,请直接写出yE与yF的大小关系;
(2)连接EF,AE.当S四边形OFEB=3S△OFE时,直接写yE与yF的大小关系及四边形OFEA的形状.
如图,在x轴上有两点A(m,0),B(n,0)(n>m>0).分别过点A,点B作x轴的垂线,交抛物线y=x2于点C、点D.直线OC交直线BD于点E,直线OD交直线AC于点F,点E、点F的纵坐标分别记为yE,yF.特例探究填空
m = 1, n = 2: yE = 2, yF = 2
m = 3, n = 5: yE = 15, yF = 15
M(m, m²), N(n, n²)
OM斜率: (m² - 0)/(m - 0) = m, 方程y = mx; 取x = n, y = mn, E(n, mn)
ON斜率: (n² - 0)/(n - 0) = n, 方程y = nx; 取x = m, y = mn, F(m, mn)
(1)
M(m, am²), N(n, an²)
OM斜率: (am² - 0)/(m - 0) = am, 方程y = amx; 取x = n, y = amn, E(n, amn)
ON斜率: (an² - 0)/(n - 0) = an, 方程y = anx; 取x = m, y = amn, F(m, amn)
yE = yF
(2)
上面已得yE = yF, 四边形OFEB为梯形, 面积S = (1/2)(FE + OB)BE = (1/2)(n - m + n)mn
S△OFE = (1/2)FE*BE = (1/2)(n - m)mn
S四边形OFEB=3S△OFE, (1/2)(n - m + n)mn = (3/2)mn(n - m)
n = 2m
EF = OA, 四边形OFEA为平行四边形