1
M是AB中点,故：OM=(OA+OB)/2
即：MP=OP-OM=xOA+yOB-(OA+OB)/2
=(x-1/2)OA+(y-1/2)OB
2
a^2(x-1/2)^2+b^2(y-1/2)^2=1
即：(x-1/2)^2/(1/a^2)+(y-1/2)^2/(1/b^2)=1
即：x和y构成一个椭圆方程
令：x=1/2+cost/a,y=1/2+sint/b
|OP|^2=x^2|OA|^2+y^2|OB|^2+2xyOA·OB
=x^2a^2+y^2b^2
=a^2(1/2+cost/a)^2+b^2(1/2+sint/b)^2
=(a^2+b^2)/4+cost^2+sint^2+acost+bsint
=2+acost+bsint
故：|OP|^2的最大值：2+√(a^2+b^2)=4
即：|OP|的最大值：2
求四边形OAPB面积这一问稍微有点问题：
因为P点可以在任意位置,构成的四边形不一定是OAPB
按照一象限的话,也可能是凸或是凹四边形
既然求最大值,就按照凸四边形作了：
SOAPB=S△OAB+S△PAB
S△OAB=ab/2
AP=OP-OA=xOA+yOB-OA=(x-1)OA+yOB
BP=OP-OB=xOA+yOB-OB=xOA+(y-1)OB
故：S△PAB=|AP×BP|=|x(x-1)OA×OA+y(y-1)OB×OB+((x-1)(y-1)-xy)OA×OB|
=|((x-1)(y-1)-xy)OA×OB|=|(1-x-y)ab|
=|1-x-y|ab
=|1-1/2-cost/a-1/2-sint/b|ab
=|cost/a+sint/b|ab
即：SOAPB=S△OAB+S△PAB
=ab/2+|cost/a+sint/b|ab
即：SOAPB的最大值：ab/2+ab√(1/a^2+1/b^2)
=2+ab/2
a^2+b^2=4≥2ab,即：ab≤2
即SOAPB的最大值：2+1=3