椭圆x^2/4+y^/2=1中,过P(1,1)的玄,刚好被p点平分,求此弦的直线方程

Question

椭圆x^2/4+y^/2=1中,过P(1,1)的玄,刚好被p点平分,求此弦的直线方程

Answer 1

设直线与椭圆交于AB二点,A(x1,y1),B(x2,y2)
则有:x1+x2=2,y1+y2=2
代入方程得:
x1^2/4+y1^2/2=1
x2^2/4+y2^2/2=1
相减得:
(x1+x2)(x1-x2)/4+(y1+y2)(y1-y2)/2=0
2(x1-x2)/4=-2(y1-y2)/2
即直线AB的斜率k=(y1-y2)/(x1-x2)=-1/2.
那么直线方程是:y-1=-1/2(x-1)
即:y=-1/2x+3/2.