题目

题型：填空题难度：一般来源：不详

已知

3

x+y

=

4

y+z

=

5

z+x

，则

x2+y2+z2

xy+yz+zx

=______．

答案

∵

3

x+y

=

4

y+z

=

5

z+x

，
∴

3 x+y = 4 y+z 3 x+y = 5 z+x 4 y+z = 5 z+x

⇒

y+z 4 = x+y 3 x+z 5 = x+y 3 x+z 5 = y+z 4

⇒

x+y

3

=

y+z

4

=

x+z

5

∴令

x+y

3

=

y+z

4

=

x+z

5

=a
则有   

x+y=3a          ① y+z=4a          ② x+z=5a          ③

  
由①+②+③得   x+y+z=6a     ④
由④-①得   z=3a，
同理解得x=2a，y=a
∴

x2+y2+z2

xy+yz+zx

=

(22+12+32)a2

(2×1+1×3+2×3)a2

=

14

11

故答案为

14

11

．