求过P(1,1)作椭圆x2/4+y2/2=1的弦长AB并使P为弦中点,求直线方程及弦AB的绝对值
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求过P(1,1)作椭圆x2/4+y2/2=1的弦长AB并使P为弦中点,求直线方程及弦AB的绝对值
求过P(1,1)作椭圆x2/4+y2/2=1的弦长AB并使P为弦中点,求直线方程及弦AB的绝对值
求过P(1,1)作椭圆x2/4+y2/2=1的弦长AB并使P为弦中点,求直线方程及弦AB的绝对值
设A,B点坐标分别为(x1,y1),(x2,y2)
P是AB的中点,所以x1+x2=y1+y2=2,
1)将A B坐标带入椭圆方程有,
x1^2/4+y1^2/2=1
x2^2/4+y2^2/2=1
将上两式相减有(x1-x2)(x1+x2)/4+(y1-y2)(y1+y2)/2=0,即
(x1-x2)/2+(y1-y2)=0,所以所在直线方程的斜率k=(y1-y2)/(x1-x2)=-1/2
所以直线方程y-1=-1/2(x-1),即x+2y-3=0
2)y=-x/2+3/2带入椭圆方程化简有3x^2-6x+1=0
x1,x2是上述二次方程的跟,所以x1*x2=1/3
|AB|=(√(1+k^2))|x1-x2|=√5/2*√((x1+x2)^2-2x1x2)=5√6/6