The number of number-pairs $(x, y)$ which will satisfy the equation $x^{2} - x y + y^{2} = 4 (x + y - 4)$ is

1

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2

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4

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None of these

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Solution

The correct option is A $1$
Given, $x^{2} - x y + y^{2} = 4 (x + y - 4)$
$\Rightarrow 2 x^{2} - 2 x y + 2 y^{2} - 8 x - 8 y + 32 = 0$
$\Rightarrow (x^{2} - 2 x y + y^{2}) + (x^{2} - 8 x + 16) + (y^{2} - 8 y + 16) = 0$
$\Rightarrow (x - y)^{2} + (x - 4)^{2} + (y - 4)^{2} = 0$
$\Rightarrow (x - y) = 0, x - 4 = 0, y - 4 = 0$
Hence, there is only one solution $(4, 4)$ .

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