作业帮椭圆X2/36+Y2/9=1上有俩动点PQ,E(3,0),EP垂直于EQ,则向量EP点乘向量QP的最小值为多少?答案是6
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椭圆X2/36+Y2/9=1上有俩动点PQ,E(3,0),EP垂直于EQ,则向量EP点乘向量QP的最小值为多少?答案是6
椭圆X2/36+Y2/9=1上有俩动点PQ,E(3,0),EP垂直于EQ,则向量EP点乘向量QP的最小值为多少?答案是6
椭圆X2/36+Y2/9=1上有俩动点PQ,E(3,0),EP垂直于EQ,则向量EP点乘向量QP的最小值为多少?答案是6
E': x^2/36 + y^2/9 =1
P,Q is on E'
E(3,0)
EP 垂直于EQ
let P (x1,y1), Q(x2,y2)
EP 垂直于EQ
=>EP.EQ=0
(x1-3,y1).(x2-3,y2)=0
(x1-3)(x2-3)+ y1y2=0
x1x2-3(x1+x2)+9 + y1y2=0
EP.QP
(x1-3,y1).(x1-x2,y1-y2)
=(x1-3)(x1-x2) + y1(y1-y2)
=x1^2-x1x2-3x1+3x2+y1^2-y1y2
=x1^2+y1^2 - (x1x2+y1y2-3(x1+x2)+9 ) +9-6x1
=x1^2+y1^2-6x1+9
d(EP.QP)/dx1 = 2x1+ 2y1(dy1/dx1)-6 (1)
x^2/36 + y^2/9 =1
(x1,y1) is on the ellipse
x1^2/36 + y1^2/9 =1
x1/18+(2y1/9)dy1/dx1=0
dy1/dx1= -x1/4y1 (2)
sub (2) into (1)
d(EP.QP)/dx1 = 2x1+ 2y1(-x1/4y1)-6
= 3x1/2-6
put d(EP.QP)/dx1 =0
=> x1=4
x1^2/36 + y1^2/9 =1
y1^2=5
min(EP.QP)=x1^2+y1^2-6x1+9 ( at x1=4 , y1^2 =5)
=16+5-24+9
=6