please-find-the-integral-solutions-x-and-y-xy-7-2-x-2-y-2-

Question Number 31763 by RAMANUJAN last updated on 14/Mar/18

p l e a s e f i n d t h e i n t e g r a l s o l u t i o n s (x a n d y)

{(x y - 7)}^{2} = x^{2} + y^{2}

Answered by MJS last updated on 14/Mar/18

(- 7, 0)

(- 4, - 3)

(- 3, - 4)

(0, - 7)

(0, 7)

(3, 4)

(4, 3)

(7, 0)

Commented by MJS last updated on 14/Mar/18

\dots now I see that

4 \times 3 - 7 = 5

4^{2} - 3^{2} = 7

3^{2} + 4^{2} = 5^{2}

these might have helped with

the construction of the given

equation

Commented by Joel578 last updated on 14/Mar/18

Sir pls explain the way . I^{'} m just guessing the

answer

Commented by MJS last updated on 14/Mar/18

I solved the equation

y = \frac{7 x \pm \sqrt{x^{4} - x^{2} + 49}}{x^{2} - 1}

\Rightarrow x^{4} - x^{2} + 49 must be a square number

then I tried \dots

Answered by Joel578 last updated on 14/Mar/18

(0, \pm 7) and (\pm 7, 0)

Answered by ajfour last updated on 14/Mar/18

x^{2} y^{2} - 14 x y + 49 = x^{2} + y^{2}

x^{2} y^{2} - 16 x y + 64 = {(x - y)}^{2} + 15

{(x y - 8)}^{2} = {(x - y)}^{2} + 15

(x y - 8 - x + y) (x y - 8 + x - y) = 15

\Rightarrow x y - 8 - x + y = 3 a n d

x y - 8 + x - y = 5

\Rightarrow x y = 12 a n d x - y = 1

\Rightarrow (4, 3) o r (- 3, - 4)

a l t e r n a t i v e l y

x y - 8 - x + y = 5 a n d

x y - 8 + x - y = 3

\Rightarrow x y = 12 a n d y - x = 1

s o (3, 4) o r (- 4, - 3)

a l t e r n a t i v e l y

x y - 8 - x + y = - 5 a n d

x y - 8 + x - y = - 3

\Rightarrow x y = 8 a n d x - y = 1

\Rightarrow n o i n t e g r a l s o l u t i o n

o r x y - 8 - x + y = - 3 a n d

x y - 8 + x - y = - 5

\Rightarrow x y = 8 a n d y - x = 1

\Rightarrow n o i n t e g r a l s o l u t i o n, a g a i n .

o t h e r w i s e

x y - 8 - x + y = 15 a n d

x y - 8 + x - y = 1

\Rightarrow x y = 16 a n d y - x = 7

\Rightarrow n o i n t e g r a l s o l u t i o n

a l t e r n a t i v e l y

x y - 8 - x + y = - 15 a n d

x y - 8 + x - y = - 1

\Rightarrow x y = 0 a n d x - y = 7

\Rightarrow (7, 0) a n d (0, - 7)

o t h e r w i s e

x y - 8 - x + y = - 1 a n d

x y - 8 + x - y = - 15

\Rightarrow x y = 0 a n d y - x = 7

\Rightarrow (0, 7) a n d (- 7, 0)

h e n c e t h e i n t e g r a l s o l u t i o n s

f o r (x, y) a r e

(4, 3), (- 3, - 4), (3, 4), (- 4, - 3),

a n d (7, 0), (0, - 7), (0, 7), (- 7, 0) .

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