Question

椭圆x方+4y方=4截直线y=x+t所得的最大弦长是多少

Answer 1

直线y=x+t代入得:

x^2+4(x+t)^2=4

x^2+4x^2+8xt+4t^2=4

5x^2+8tx+4t^2-4=0

x1+x2=-8t/5

x1x2=(4t^2-4)/5.

|x1-x2|^2=(x1+x2)^2-4x1x2=64t^2/25-16(t^2-1)/5=16(5-t^2)/25

设弦长是L

则L=根号(1+k^2)*|x1-x2|=根号(1+1)*|x1-x2|

即L^2=2*16(5-t^2)/25

当t=0时,L^2有最大值是L^2=32*5/25=32/5

即最大弦长是L=4根号10/5