设F1,F2,分别是椭圆C:(x^2/a^2)+(y^2/b^2)=1(a>b>0)的左,右焦点,过F1且斜率为1的直线l与C交于A,B两点且|AF2|、|AB|、|BF2|成等差数列 (1) 求证a=根号2*b(2) 设点P(0,1)满足|PA|=|PB|,求椭圆C方程
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![设F1,F2,分别是椭圆C:(x^2/a^2)+(y^2/b^2)=1(a>b>0)的左,右焦点,过F1且斜率为1的直线l与C交于A,B两点且|AF2|、|AB|、|BF2|成等差数列 (1) 求证a=根号2*b(2) 设点P(0,1)满足|PA|=|PB|,求椭圆C方程](/uploads/image/z/1592337-57-7.jpg?t=%E8%AE%BEF1%2CF2%2C%E5%88%86%E5%88%AB%E6%98%AF%E6%A4%AD%E5%9C%86C%EF%BC%9A%28x%5E2%2Fa%5E2%29%2B%28y%5E2%2Fb%5E2%29%3D1%28a%3Eb%3E0%29%E7%9A%84%E5%B7%A6%2C%E5%8F%B3%E7%84%A6%E7%82%B9%2C%E8%BF%87F1%E4%B8%94%E6%96%9C%E7%8E%87%E4%B8%BA1%E7%9A%84%E7%9B%B4%E7%BA%BFl%E4%B8%8EC%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%E4%B8%94%7CAF2%7C%E3%80%81%7CAB%7C%E3%80%81%7CBF2%7C%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97+%EF%BC%881%EF%BC%89+%E6%B1%82%E8%AF%81a%3D%E6%A0%B9%E5%8F%B72%2Ab%EF%BC%882%EF%BC%89+%E8%AE%BE%E7%82%B9P%280%2C1%29%E6%BB%A1%E8%B6%B3%7CPA%7C%3D%7CPB%7C%2C%E6%B1%82%E6%A4%AD%E5%9C%86C%E6%96%B9%E7%A8%8B)
设F1,F2,分别是椭圆C:(x^2/a^2)+(y^2/b^2)=1(a>b>0)的左,右焦点,过F1且斜率为1的直线l与C交于A,B两点且|AF2|、|AB|、|BF2|成等差数列 (1) 求证a=根号2*b(2) 设点P(0,1)满足|PA|=|PB|,求椭圆C方程
设F1,F2,分别是椭圆C:(x^2/a^2)+(y^2/b^2)=1(a>b>0)的左,右焦点,过F1且斜率为1的直线l与C交于A,B两点
且|AF2|、|AB|、|BF2|成等差数列
(1) 求证a=根号2*b
(2) 设点P(0,1)满足|PA|=|PB|,求椭圆C方程
设F1,F2,分别是椭圆C:(x^2/a^2)+(y^2/b^2)=1(a>b>0)的左,右焦点,过F1且斜率为1的直线l与C交于A,B两点且|AF2|、|AB|、|BF2|成等差数列 (1) 求证a=根号2*b(2) 设点P(0,1)满足|PA|=|PB|,求椭圆C方程
1.设A(x1,y1),B(x2,y2),左焦点(-c,0),直线l:y=x+c
由题意得|AF2|+|BF2|=2|AB|
∵ |AF1|+|AF2|=2a. .①
|BF1|+|BF2|=2a.②
①+②得(|AF1|+|BF1|)+(|AF2|+|BF2|)=4a
即|AB|+2|AB|=4a,|AB|=4a/3……③
直线y=x+c代入椭圆方程x²/a² + y²/b² =1,得(a²+b²)x²+2a²cx+a²c²-a²b²=0
x1+x2=-2a²c/(a²+b²),x1*x2=(a²c²-a²b²)/(a²+b²)
|AB|=√(1+k²)|x1-x2|=√2*√[(x1+x2)²-4x1x2]
把c²=a²-b²代入,得|AB|=4ab²/(a²+b²)……④
由③④得a²=2b²,a=√2b
(2)设点P(0,1)满足|PA|=|PB|,则|PA|²=|PB|²
所以(x1+1)²+y1²=(x2+1)²+y2²
(x1+1)²-(x2+1)²+y1²-y2²=0
(x1-x2)(x1+x2+2) + (y1-y2)(y1+y2)=0
(x1-x2)(x1+x2+2) + [(x1+c)-(x2+c)][(x1+c)+(x2+c)]=0
(x1-x2)(x1+x2+2) + (x1-x2)(x1+x2+2c)=0
2(x1-x2)[(x1+x2)+1+c]=0
∵x1≠x2,∴x1-x2≠0,∴(x1+x2)+1+c=0
即-2a²c/(a²+b²)+1+c=0
∵a²=2b²,c²=a²-b²,得到b=c
代入上式整理得b=3,a=18
椭圆C方程x²/18+y²/9=1