2.5　直线与圆锥曲线

课时过关·能力提升

1.若椭圆 x^2/36+y^2/9=1的弦被点(4,2)平分,则此弦所在直线的斜率为(　　)

A.2 B.-2 C. 1/3 D.-1/2

解析:设弦两端点A(x1,y1),B(x2,y2),则x1+x2=8,y1+y2=4,

　　又{■((x_1^2)/36+(y_1^2)/9=1",①" @(x_2^2)/36+(y_2^2)/9=1".②" )┤

　　①-②得

　　("(" x_1+x_2 ")(" x_1 "-" x_2 ")" )/36+("(" y_1+y_2 ")(" y_1 "-" y_2 ")" )/9=0,

　　即 (8"(" x_1 "-" x_2 ")" )/36+(4"(" y_1 "-" y_2 ")" )/9=0,

　　所以所求直线的斜率为 (y_1 "-" y_2)/(x_1 "-" x_2 )=-1/2.

答案:D

2.已知椭圆x2+2y2=4,则以(1,1)为中点的弦的长度为0(　　)

A.3√2 B.2√3

C.√30/3 D.3/2 √6

解析:依题设弦的端点为A(x1,y1),B(x2,y2),

　　则x1+x2=2,y1+y2=2,

　　又x_1^2+2y_1^2=4,x_2^2+2y_2^2=4,

　　所以x_1^2-x_2^2=-2(y_1^2-y_2^2),

　　此弦的斜率k=(y_1 "-" y_2)/(x_1 "-" x_2 )=-(x_1+x_2)/(2"(" y_1+y_2 ")" )=-1/2,

　　所以此弦所在的直线方程为y-1=-1/2(x-1),

即y=-1/2 x+3/2.