Question

已知x(y+z)/27=y(x+z)/32=z(x+y)/35,求z^2/x^2+y^2

Answer 1

令x(y+z)/27=y(x+z)/32=z(x+y)/35=k

xy+xz=27k

xy+yz=32k

xz+yz=35k

相加

2(xy+yz+zx)=94k

xy+yz+zx=47k

分别减去xy+xz=27k

xy+yz=32k

xz+yz=35k

yz=20k,zx=15k,xy=12k

y=20k/z,x=15k/z

xy=300k^2/z^2=12k

25k/z^2=1

z^2=25k

y=20k/z,

y^2=400k^2/z^2=16k

x=15k/z,

x^2=225k^2/z^2=9k

所以z^2/(x^2+y^2)

=(25k)/(16k+9k)

=1