题目内容
已知点(1,1)是椭圆
+
=1某条弦的中点,则此弦所在的直线方程为:______.
x2 |
4 |
y2 |
2 |
设以A(1,1)为中点椭圆的弦与椭圆交于E(x1,y1),F(x2,y2),
∵A(1,1)为EF中点,
∴x1+x2=2,y1+y2=2,
把E(x1,y1),F(x2,y2)分别代入椭圆
+
=1,
可得
+
=1,
+
=1
两式相减,可得(x1+x2)(x1-x2)+2(y1+y2)(y1-y2)=0,
∴2(x1-x2)+4(y1-y2)=0,
∴k=
=-
∴以A(1,1)为中点椭圆的弦所在的直线方程为:y-1=-
(x-1),
整理,得x+2y-3=0.
故答案为:x+2y-3=0.
∵A(1,1)为EF中点,
∴x1+x2=2,y1+y2=2,
把E(x1,y1),F(x2,y2)分别代入椭圆
x2 |
4 |
y2 |
2 |
可得
x12 |
4 |
y12 |
2 |
x22 |
4 |
y22 |
2 |
两式相减,可得(x1+x2)(x1-x2)+2(y1+y2)(y1-y2)=0,
∴2(x1-x2)+4(y1-y2)=0,
∴k=
y1-y2 |
x1-x2 |
1 |
2 |
∴以A(1,1)为中点椭圆的弦所在的直线方程为:y-1=-
1 |
2 |
整理,得x+2y-3=0.
故答案为:x+2y-3=0.
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