(x^2 /(2^2 −1^2 ))+(y^2 /(2^2 −3^2 ))+(z^2 /(2^2 −5^2 ))+(w^2 /(2^2 −7^2 ))=1 (x^2 /(4^2 −1^2 ))+(y^2 /(4^2 −3^2 ))+(z^2 /(4^2 −5^2 ))+(w^2 /(4^2 −7^2 ))=1 (x^2 /(6^2 −1^2 ))+(y^2 /(6^2 −3^2 ))+(z^2 /(6^2 −5^2 ))+(w^2 /(6^2 −7^2 ))=1 (x^2 /(8^2 −1^2 ))+(y^2 /(8^2 −3^2 ))+(z^2 /(8^2 −5^2 ))+(w^2 /(8^2 −7^2 ))=1 find (x^2 +y^2 +z^2 +w^2 ).


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Question Number 193182 by York12 last updated on 06/Jun/23

$\frac{x^{2}}{2^{2} - 1^{2}} + \frac{y^{2}}{2^{2} - 3^{2}} + \frac{z^{2}}{2^{2} - 5^{2}} + \frac{w^{2}}{2^{2} - 7^{2}} = 1$ $\frac{x^{2}}{4^{2} - 1^{2}} + \frac{y^{2}}{4^{2} - 3^{2}} + \frac{z^{2}}{4^{2} - 5^{2}} + \frac{w^{2}}{4^{2} - 7^{2}} = 1$ $\frac{x^{2}}{6^{2} - 1^{2}} + \frac{y^{2}}{6^{2} - 3^{2}} + \frac{z^{2}}{6^{2} - 5^{2}} + \frac{w^{2}}{6^{2} - 7^{2}} = 1$ $\frac{x^{2}}{8^{2} - 1^{2}} + \frac{y^{2}}{8^{2} - 3^{2}} + \frac{z^{2}}{8^{2} - 5^{2}} + \frac{w^{2}}{8^{2} - 7^{2}} = 1$ $f i n d (x^{2} + y^{2} + z^{2} + w^{2}) .$
	Answered by York12 last updated on 06/Jun/23
	$Let s takes the values (4 , 16 , 36 , 64) ∴ (x^2 /(s−1))+(y^2 /(s−9))+(z^2 /(s−25))+(w^2 /(s−49))=1 ⇒ x^2 (s−9)(s−25)(s−49) + y^2 (s−1)(s−25)(s−49)+z^2 (s−1)(s−9)(s−49)+w^2 (s−1)(s−9)(s−25)=(s−1)(s−9)(s−25)(s−49) ⇒x^2 (s−9)(s−25)(s−49) + y^2 (s−1)(s−25)(s−49)+z^2 (s−1)(s−9)(s−49)+w^2 (s−1)(s−9)(s−25)−(s−1)(s−9)(s−25)(s−49)=0 let the previous expression be a quartic polynomial in s with leading coefficient of (−1) and we know the values of s for which that holds s ∈ { 4 , 16 , 36 , 64 } Then : x^2 (s−9)(s−25)(s−49) + y^2 (s−1)(s−25)(s−49)+z^2 (s−1)(s−9)(s−49)+w^2 (s−1)(s−9)(s−25)−(s−1)(s−9)(s−25)(s−49)= −(s−4)(s−16)(s−36)(s−64) = f(s) f(1)= −(3×15×35×63)=−x^2 (8×24×48) ⇒x^2 =((3×15×35×63)/(8×24×48)) .......(i) f(9)=(5×7×27×55)=y^2 (8×16×40) ⇒ y^2 =(((5×7×27×55))/((8×16×40))) .......(ii) f(25)=−(21×9×11×39)=−z^2 (24×16×24) ⇒ z^2 =(((21×9×11×39))/((24×16×24)))......(iii) f(49)=(45×33×13×15)=w^2 (48×40×24) ⇒w^2 =(((45×33×13×15))/((48×40×24))) .......(iv) (i)+(ii)+(iii)+(iv)= 36 ★$
	$L e t s t a k e s t h e v a l u e s (4, 16, 36, 64)$ $∴ \frac{x^{2}}{s - 1} + \frac{y^{2}}{s - 9} + \frac{z^{2}}{s - 25} + \frac{w^{2}}{s - 49} = 1$ $\Rightarrow x^{2} (s - 9) (s - 25) (s - 49) + y^{2} (s - 1) (s - 25) (s - 49) + z^{2} (s - 1) (s - 9) (s - 49) + w^{2} (s - 1) (s - 9) (s - 25) = (s - 1) (s - 9) (s - 25) (s - 49)$ $\Rightarrow x^{2} (s - 9) (s - 25) (s - 49) + y^{2} (s - 1) (s - 25) (s - 49) + z^{2} (s - 1) (s - 9) (s - 49) + w^{2} (s - 1) (s - 9) (s - 25) - (s - 1) (s - 9) (s - 25) (s - 49) = 0$ $let the previous expression be a quartic$ $p o l y n o m i a l i n s w i t h l e a d i n g c o e f f i c i e n t o f (- 1)$ $a n d w e k n o w t h e v a l u e s o f s f o r w h i c h t h a t$ $h o l d s s \in {4, 16, 36, 64}$ $T h e n :$ $x^{2} (s - 9) (s - 25) (s - 49) + y^{2} (s - 1) (s - 25) (s - 49) + z^{2} (s - 1) (s - 9) (s - 49) + w^{2} (s - 1) (s - 9) (s - 25) - (s - 1) (s - 9) (s - 25) (s - 49) =$ $- (s - 4) (s - 16) (s - 36) (s - 64) = f (s)$ $f (1) = - (3 \times 15 \times 35 \times 63) = - x^{2} (8 \times 24 \times 48)$ $\Rightarrow x^{2} = \frac{3 \times 15 \times 35 \times 63}{8 \times 24 \times 48} . . . . . . . (i)$ $f (9) = (5 \times 7 \times 27 \times 55) = y^{2} (8 \times 16 \times 40)$ $\Rightarrow y^{2} = \frac{(5 \times 7 \times 27 \times 55)}{(8 \times 16 \times 40)} . . . . . . . (i i)$ $f (25) = - (21 \times 9 \times 11 \times 39) = - z^{2} (24 \times 16 \times 24)$ $\Rightarrow z^{2} = \frac{(21 \times 9 \times 11 \times 39)}{(24 \times 16 \times 24)} . . . . . . (i i i)$ $f (49) = (45 \times 33 \times 13 \times 15) = w^{2} (48 \times 40 \times 24)$ $\Rightarrow w^{2} = \frac{(45 \times 33 \times 13 \times 15)}{(48 \times 40 \times 24)} . . . . . . . (i v)$ $(i) + (i i) + (i i i) + (i v) = 36 ★$
	Commented by manxsol last updated on 06/Jun/23

	$d o m i n i o d e f (s) = {4, 16, 36, 64}$ $e v a l u a t i o n = {1, 9, 25, 49}$ $h o w d o y o u e x p l a n a t i o n ?$
	Commented by York12 last updated on 07/Jun/23
	${4,16,36,64} that is the solution set for which f(s)=0 {1,9,25,49} those are just values I have plugged them into the function to eliminate x^2 , y^2 , z^2 , w^2 so that I can obtain their values$
	${4, 16, 36, 64} t h a t i s t h e s o l u t i o n s e t$ $f o r w h i c h f (s) = 0$ ${1, 9, 25, 49} t h o s e a r e j u s t v a l u e s I h a v e$ $p l u g g e d t h e m i n t o t h e f u n c t i o n t o$ $e l i m i n a t e x^{2}, y^{2}, z^{2}, w^{2}$ $s o t h a t I c a n o b t a i n t h e i r v a l u e s$
	Commented by manxsol last updated on 07/Jun/23

	$t h a n k f o r e x p l a n a t i o n .$
	Commented by York12 last updated on 07/Jun/23

	$y o u a r e w e l c o m e$
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