已知A(-2,-2)B(-2,6)C(4,-2),P点在圆x2+y2=4上运动,求PA2+PB2+PC2的最大值和最小值
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![已知A(-2,-2)B(-2,6)C(4,-2),P点在圆x2+y2=4上运动,求PA2+PB2+PC2的最大值和最小值](/uploads/image/z/8677236-12-6.jpg?t=%E5%B7%B2%E7%9F%A5A%28-2%2C-2%29B%28-2%2C6%29C%284%2C-2%29%2CP%E7%82%B9%E5%9C%A8%E5%9C%86x2%2By2%3D4%E4%B8%8A%E8%BF%90%E5%8A%A8%2C%E6%B1%82PA2%2BPB2%2BPC2%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC)
已知A(-2,-2)B(-2,6)C(4,-2),P点在圆x2+y2=4上运动,求PA2+PB2+PC2的最大值和最小值
已知A(-2,-2)B(-2,6)C(4,-2),P点在圆x2+y2=4上运动,求PA2+PB2+PC2的最大值和最小值
已知A(-2,-2)B(-2,6)C(4,-2),P点在圆x2+y2=4上运动,求PA2+PB2+PC2的最大值和最小值
设P点为(x,y),则
f(x)=PA²+PB²+PC²=(x+2)²+(y+2)²+(x+2)²+(y-6)²+(x-4)²+(y+2)²=3x²+3y²-4y+68
因为x²+y²=4,所以x²=4-y²
f(x)=12-3y²+3y²-4y+68=80-4y,-2≤y≤2
当y=-2,f(x)有最大值88
当y=2,f(x)有最小值72
(PA)^2=(x+2)^2+(y+2)^2=x^2+y^2+4x+4y+8, 由于P在圆x^2+y^2=4上
所以,(PA)^2=4x+4y+12 同理,
(PB)^2=(x+2)^2+(y-6)^2=x^2+y^2+4x-12y+40=4x-12y+44
(PC)^2=(x-4)^2+(y+2)^2=-8x+4y+24
(PA)^2+(PB)^2+(PC)^2<...
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(PA)^2=(x+2)^2+(y+2)^2=x^2+y^2+4x+4y+8, 由于P在圆x^2+y^2=4上
所以,(PA)^2=4x+4y+12 同理,
(PB)^2=(x+2)^2+(y-6)^2=x^2+y^2+4x-12y+40=4x-12y+44
(PC)^2=(x-4)^2+(y+2)^2=-8x+4y+24
(PA)^2+(PB)^2+(PC)^2
=- 4y+80
因为 -2≤y≤2
所以,
(PA)^2+(PB)^2+(PC)^2 的最大值是88 .最小值是 72
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