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Question Number 82071 by jagoll last updated on 18/Feb/20

x≠ y ≠z ≠ 0 xy + xz + yz = 0 prove that ((x+y)/z)+((x+z)/y)+((y+z)/x) = −3

x≠y≠z≠0xy+xz+yz=0provethatx+yz+x+zy+y+zx=−3

Answered by TANMAY PANACEA last updated on 18/Feb/20

((x+y)/z)+((x+z)/y)+((y+z)/x)+3−3 ((x+y+z)/z)+((x+y+z)/y)+((x+y+z)/x)−3 (x+y+z)((1/x)+(1/y)+(1/z))−3 (x+y+z)(yz+zx+xy)×(1/(xyz))−3 0−3=−3

x+yz+x+zy+y+zx+3−3x+y+zz+x+y+zy+x+y+zx−3(x+y+z)(1x+1y+1z)−3(x+y+z)(yz+zx+xy)×1xyz−30−3=−3

Commented by jagoll last updated on 18/Feb/20

thank you sir

thankyousir

Answered by MJS last updated on 18/Feb/20

xy+xz+yz=0 ⇒ z=−((xy)/(x+y)) ((x+y)/(−((xy)/(x+y))))+((x−((xy)/(x+y)))/y)+((y−((xy)/(x+y)))/x)= =−(((x+y)^2 )/(xy))+(x^2 /((x+y)y))+(y^2 /(x(x+y)))= =((−(x+y)^3 +x^3 +y^3 )/(x(x+y)y))=((−3x^2 y−3xy^2 )/(x(x+y)y))= =((−3x(x+y)y)/(x(x+y)y))=−3

xy+xz+yz=0⇒z=−xyx+yx+y−xyx+y+x−xyx+yy+y−xyx+yx==−(x+y)2xy+x2(x+y)y+y2x(x+y)==−(x+y)3+x3+y3x(x+y)y=−3x2y−3xy2x(x+y)y==−3x(x+y)yx(x+y)y=−3

Commented by jagoll last updated on 18/Feb/20

thank you mister

thankyoumister

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