x^2 −yz=a^n y^2 −zx=b^n z^2 −xy=c^n find (x , y , z) in terms of (a , b , c)


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Question Number 192875 by York12 last updated on 30/May/23

$x^{2} - y z = a^{n}$ $y^{2} - z x = b^{n}$ $z^{2} - x y = c^{n}$ $f i n d (x, y, z) i n t e r m s o f (a, b, c)$
	Answered by Frix last updated on 30/May/23

	$My path :$ $1 . y = p x \land z = q x$ $2 . p = u + v \land q = u - v$ ${It}^{'} s then possible to solve the system .$ $I got$ $u = \frac{b^{2 n} + c^{2 n} - a^{n} (b^{n} + c^{n})}{2 (a^{2 n} - b^{n} c^{n})} \land v = \frac{(a^{n} + b^{n} + c^{n}) (c^{n} - b^{n})}{2 (a^{2 n} - b^{n} c^{n})}$ $p = \frac{b^{2 n} - a^{n} c^{n}}{a^{2 n} - b^{n} c^{n}} \land q = \frac{c^{2 n} - a^{n} b^{n}}{a^{2 n} - b^{n} c^{n}}$ $x = \pm \frac{a^{2 n} - b^{n} c^{n}}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$ $y = \pm \frac{b^{2 n} - a^{n} c^{n}}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$ $z = \pm \frac{c^{2 n} - a^{n} b^{n}}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$ $I^{'} m too lazy to type the complete workings$ $but everybody should be able to do it if$ $needed .$
	Commented by York12 last updated on 30/May/23

	$t h a n k s s o m u c h s i r t h a t w a s m o r e t h a n$ $e n o u g h$
	Commented by York12 last updated on 30/May/23

	$y o u h a v e e x p l o i t e d t h a t a l l o f t h e m a r e$ $h o m o g e n e o u s e q u a t i o n s o f t h e s e c o n d d e g r e e$
	Answered by MM42 last updated on 30/May/23

	$1) x^{2} - y z = a^{n} \overset{\times y}{\to} x^{2} y - y^{2} z = a^{n} y (i)$ $2) y^{2} - z x = b^{n} \overset{\times z}{\to} y^{2} z - z^{2} x = b^{n} z (i i)$ $3) z^{2} - x y = c^{n} \overset{\times x}{\to} {x z}^{2} - x^{2} y = c^{n} x (i i i)$ $(i) + (i i) + (i i i) \to c^{n} x + a^{n} y + b^{n} z = 0 (i v)$ $i f 1) \overset{\times z}{\to} & 2) \overset{\times x}{\to} & 3) \overset{\times y}{\to} a n d s u m o f o b t i n e d r e l a t i o n \to b^{n} x + c^{n} y + a^{n} z = 0 (v)$ $a^{n} \times (i v) - b^{n} \times (v) \to (a^{n} c^{n} - b^{2 n}) x + (a^{2 n} - b^{n} c^{n}) y = 0$ $\Rightarrow x = \frac{a^{2 n} - b^{n} c^{n}}{b^{2 n} - a^{n} c^{n}} y (v i)$ $c^{n} \times (i v) - a^{n} \times (v) \to (c^{2 n} - a^{n} b^{n}) x + (b^{n} c^{n} - a^{2 n}) z = 0$ $\Rightarrow x = \frac{a^{2 n} - b^{n} c^{n}}{a^{n} b^{n} - c^{2 n}} z (v i i)$ $(v i) & (v i i) \Rightarrow \frac{x}{a^{2 n} - b^{n} c^{n}} = \frac{y}{b^{2 n} - a^{n} c^{n}} = \frac{z}{a^{n} b^{n} - c^{2 n}}$
	Commented by York12 last updated on 30/May/23

	$g o o d j o b s i r$
	Commented by Frix last updated on 30/May/23

	$But the job is not done yet . . .$
	Commented by York12 last updated on 30/May/23

	$t h e r e s t i s t r i v i a l s i r$
	Commented by York12 last updated on 30/May/23

	$1) x^{2} - y z = a^{n} \overset{\times y}{\to} x^{2} y - y^{2} z = a^{n} y (i)$ $2) y^{2} - z x = b^{n} \overset{\times z}{\to} y^{2} z - z^{2} x = b^{n} z (i i)$ $3) z^{2} - x y = c^{n} \overset{\times x}{\to} {x z}^{2} - x^{2} y = c^{n} x (i i i)$ $(i) + (i i) + (i i i) \to c^{n} x + a^{n} y + b^{n} z = 0 (i v)$ $i f 1) \overset{\times z}{\to} & 2) \overset{\times x}{\to} & 3) \overset{\times y}{\to} a n d s u m o f o b t i n e d r e l a t i o n \to b^{n} x + c^{n} y + a^{n} z = 0 (v)$ $a^{n} \times (i v) - b^{n} \times (v) \to (a^{n} c^{n} - b^{2 n}) x + (a^{2 n} - b^{n} c^{n}) y = 0$ $\Rightarrow x = \frac{a^{2 n} - b^{n} c^{n}}{b^{2 n} - a^{n} c^{n}} y (v i)$ $c^{n} \times (i v) - a^{n} \times (v) \to (c^{2 n} - a^{n} b^{n}) x + (b^{n} c^{n} - a^{2 n}) z = 0$ $\Rightarrow x = \frac{a^{2 n} - b^{n} c^{n}}{a^{n} b^{n} - c^{2 n}} z (v i i)$ $(v i) & (v i i) \Rightarrow \frac{x}{a^{2 n} - b^{n} c^{n}} = \frac{y}{b^{2 n} - a^{n} c^{n}} = \frac{z}{a^{n} b^{n} - c^{2 n}} = λ$ $b y s u b t i t u t i n g$ $w e g e t$ $λ^{2} (a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}) = 1$ $λ = \frac{\underline{+} 1}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$ $∴ x = \frac{\underline{+} (a^{2 n} - b^{n} c^{n})}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}, y = \frac{\underline{+} (b^{2 n} - c^{n} a^{n})}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$ $z = \frac{\underline{+} (c^{2 n} - a^{n} b^{n})}{\sqrt{a^{3 n} + b^{3 n} + c^{3 n} - 3 a^{n} b^{n} c^{n}}}$
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