How many ordered pairs of positive integers x - y satisfy the equation x - y - y - x -2006 xy -2006 x -2006 y -2006-0 -correct answer -3- wrong answer 0

X√(y)+y√(x)+√(2006xy) √(2006x) √(2006y) 2006=0 ⇒(√(x)+√(y)+√(2006))(√(xy) √(2006))=0 ⇒(√(xy) √(2006))=0 (∵√(x)+√(y)+√(2006) gt;0) ⇒ xy=2006=17× 59× 2 Thu ...

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Standard XII

Mathematics

Properties of Iota

How many orde...

Question

How many ordered pairs of positive integers $(x, y)$ satisfy the equation $x \sqrt{y} + y \sqrt{x} + \sqrt{2006 x y} - \sqrt{2006 x} - \sqrt{2006 y} - 2006 = 0$ ?
(correct answer + 3, wrong answer 0)

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Solution

$x \sqrt{y} + y \sqrt{x} + \sqrt{2006 x y} - \sqrt{2006 x} - \sqrt{2006 y} - 2006 = 0$
$\Rightarrow (\sqrt{x} + \sqrt{y} + \sqrt{2006}) (\sqrt{x y} - \sqrt{2006}) = 0$
$\Rightarrow (\sqrt{x y} - \sqrt{2006}) = 0 (∵ \sqrt{x} + \sqrt{y} + \sqrt{2006} > 0)$
$\Rightarrow x y = 2006 = 17 \times 59 \times 2$
Thus the solutions are
$(1, 2006), (2006, 1), (2, 1003), (1003, 2), (34, 59), (59, 34), (17, 118), (118, 17)$

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How many ordered pairs of positive integers (x,y) satisfy the equation x√y+y√x+√2006xy−√2006x−√2006y−2006=0? (correct answer + 3, wrong answer 0)

x√y+y√x+√2006xy−√2006x−√2006y−2006=0 ⇒(√x+√y+√2006)(√xy−√2006)=0 ⇒(√xy−√2006)=0 (∵√x+√y+√2006>0) ⇒xy=2006=17×59×2 Thus the solutions are (1,2006),(2006,1),(2,1003),(1003,2),(34,59),(59,34),(17,118),(118,17)

How many ordered pairs of positive integers $(x, y)$ satisfy the equation $x \sqrt{y} + y \sqrt{x} + \sqrt{2006 x y} - \sqrt{2006 x} - \sqrt{2006 y} - 2006 = 0$ ?
(correct answer + 3, wrong answer 0)