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

l e t a, b, c, x, y a n d z b e c o m p l e x n u m b e r

s u c h t h a t a = \frac{b + c}{x - 2}, b = \frac{c + a}{y - 2} c = \frac{a + b}{z - 2} .

x y + y z + z x = 1000 a n d x + y + z = 2016

f i n d t h e v a l u e o f x y z .

Answered by ajfour last updated on 03/Dec/17

Π (x - 2) = x y z - 2 Σ x y + 4 Σ x - 8

\frac{(a + b) (b + c) (c + a)}{a b c} = x y z - 2000

+ 8064 - 8

\Rightarrow x y z = \frac{(a + b) (b + c) (c + a)}{a b c} - 6056

F u r t h e r,

x + y + z = 8 + Σ (\frac{b + c}{a}) = 2016 \dots (i)

Σ (x - 2) (y - 2) = Σ (\frac{b + c}{a}) (\frac{c + a}{b})

Σ x y - 2 Σ (x + y) + 12 = Σ (\frac{b + c}{a}) (\frac{c + a}{b})

1000 - 4 \times 2016 + 12 = Σ (\frac{b + c}{a}) (\frac{c + a}{b})

\Rightarrow Σ (\frac{b + c}{a}) (\frac{c + a}{b}) = 7056

s o \frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c} a r e r o o t s o f

c u b i c e q u a t i o n :

q^{3} - 2008 q^{2} + 7056 q - (x y z + 6056) = 0

\dots . m a y c o n t i n u e . .

Commented by jota+ last updated on 03/Dec/17

T h e e n d i s

x y z = \frac{(a + b) (b + c) (c + a)}{a b c} - 6056 .

Facebook Tweet Pin