Solve-x-2-a-y-z-2-y-2-b-z-x-2-z-2-c-x-y-2-

Question Number 215356 by Abdullahrussell last updated on 04/Jan/25

S o l v e :

x^{2} = a + {(y - z)}^{2}

y^{2} = b + {(z - x)}^{2}

z^{2} = c + {(x - y)}^{2}

Answered by mr W last updated on 04/Jan/25

(x + y - z) (x - y + z) = a \dots (i)

(- x + y + z) (x + y - z) = b \dots (i i)

(x - y + z) (- x + y + z) = c \dots (i i i)

\Rightarrow {(- x + y + z)}^{2} {(x - y + z)}^{2} {(x + y - z)}^{2} = a b c

a s s u m e a b c ⩾ 0, o t h e r w i s e n o s o l u t i o n .

(- x + y + z) (x - y + z) (x + y - z) = \pm \sqrt{a b c}

\Rightarrow - x + y + z = \pm \frac{\sqrt{a b c}}{a} \dots (I)

\Rightarrow x - y + z = \pm \frac{\sqrt{a b c}}{b} \dots (I I)

\Rightarrow x + y - z = \pm \frac{\sqrt{a b c}}{c} \dots (I I I)

(I I) + (I I I) :

\Rightarrow 2 x = \pm (\frac{1}{b} + \frac{1}{c}) \sqrt{a b c}

x = \pm \frac{1}{2} (\frac{1}{b} + \frac{1}{c}) \sqrt{a b c}

s i m i l a r l y

y = \pm \frac{1}{2} (\frac{1}{c} + \frac{1}{a}) \sqrt{a b c}

z = \pm \frac{1}{2} (\frac{1}{a} + \frac{1}{b}) \sqrt{a b c}

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