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let-a-b-c-x-y-and-z-be-complex-number-such-that-a-b-c-x-2-b-c-a-y-2-c-a-b-z-2-xy-yz-zx-1000-and-x-y-z-2016-find-the-value-of-xyz-




Question Number 25066 by Mr easy last updated on 02/Dec/17
let a,b,c,x,y and z be complex number  such that a=((b+c)/(x−2)) ,b=((c+a)/(y−2))    c=((a+b)/(z−2)).  xy +yz +zx=1000 and x+y+z=2016  find the value of xyz.
leta,b,c,x,yandzbecomplexnumbersuchthata=b+cx2,b=c+ay2c=a+bz2.xy+yz+zx=1000andx+y+z=2016findthevalueofxyz.
Answered by ajfour last updated on 03/Dec/17
Π(x−2)=xyz−2Σxy+4Σx−8  (((a+b)(b+c)(c+a))/(abc))=xyz−2000                                          +8064−8  ⇒ xyz=(((a+b)(b+c)(c+a))/(abc))−6056  Further,  x+y+z=8+Σ(((b+c)/a))=2016  ...(i)  Σ(x−2)(y−2)=Σ(((b+c)/a))(((c+a)/b))  Σxy−2Σ(x+y)+12=Σ(((b+c)/a))(((c+a)/b))  1000−4×2016+12=Σ(((b+c)/a))(((c+a)/b))  ⇒ Σ(((b+c)/a))(((c+a)/b))=7056  so ((b+c)/a), ((c+a)/b), ((a+b)/c)  are roots of  cubic equation:  q^3 −2008q^2 +7056q−(xyz+6056)=0  ....may  continue..
Π(x2)=xyz2Σxy+4Σx8(a+b)(b+c)(c+a)abc=xyz2000+80648xyz=(a+b)(b+c)(c+a)abc6056Further,x+y+z=8+Σ(b+ca)=2016(i)Σ(x2)(y2)=Σ(b+ca)(c+ab)Σxy2Σ(x+y)+12=Σ(b+ca)(c+ab)10004×2016+12=Σ(b+ca)(c+ab)Σ(b+ca)(c+ab)=7056sob+ca,c+ab,a+bcarerootsofcubicequation:q32008q2+7056q(xyz+6056)=0.maycontinue..
Commented by jota+ last updated on 03/Dec/17
The end is  xyz=(((a+b)(b+c)(c+a))/(abc)) −6056.
Theendisxyz=(a+b)(b+c)(c+a)abc6056.

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