(1) A(0,4) B(3,0) C(4.2) D(x1,y1)
|AB|=√((0-3)^2+(4-0)^2)=5
kAB=(4-0)/(0-3)=-4/3
平行四边形：|CD|=|AB| kCD=kAB
√((x1-4)^2+(y1-2)^2)=5
(x1-4)^2+(y1-2)^2=25.(1)
(y1-2)/(x1-4)=-4/3
y1-2=-4/3(x1-4).(2)
(2)代入(1)：(x1-4)^2+(-4/3(x1-4))^2=25
(x1-4)^2(1+16/9)=25
(x1-4)^2=9
x1-4=±3
x1=4+3=7
y1-2=-4/3(7-4)
y1=-2
或者x1=4-3=1
y1-2=-4/3(1-4)
y1=6
D(7,-2)或者D(1,6)
(2) C(4,2)在y=k/x上
2=k/4
k=8
y=8/x
设Q(q,8/q)到AB的距离为d (q>0)
AB方程：(y-0)/(x-3)=(4-0)/(0-3)
4x+3y+12=0
d=|4q+3*8/q+12|/√(4^2+3^2)
=4|q+6/q+3|/5
SΔABQ=1/2|AB|*4|q+6/q+3|/5
=2/5*5|q+6/q+3|
=2|q+6/q+3|
q+6/q>=2√q*√(6/q)=2√6
当q=6/q时,q+6/q有最小值：2√6
SΔABQmin=2|2√6+3|=4√6+6
(2) 解法2：
C(4,2)在y=k/x上
2=k/4
k=8
y=8/x
当y=8/x上某点的切线与AB平行时,三角形ABQ的面积最小.
y'=-8/x^2=kAB=-4/3
x^2=6
x=±√6(∵在第一象限,∴x=√6)
y=8/√6=4√6/3
Q(√6,4√6/3)
AB方程：(y-0)/(x-3)=(4-0)/(0-3)
4x+3y+12=0
Q到AB的距离：d=|4*√6+3*4√6/3+12|/√(4^2+3^2)
=(8√6+12)/5
SΔABQmin=1/2*|AB|*d
=1/2*5*(8√6+12)/5
=4√6+6