The number of points (a, b), where a and b are positive integers, lying on the hyperbola $x^{2} - y^{2} = 512$ is

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Solution

The correct option is B 4
$a^{2} - b^{2} = 512 \Rightarrow (a + b) (a - b) = 2^{9}$
Since $a$ and $b$ are positive integers, $a + b > 0$ . Hence $a - b$ must be $> 0$ .
Therefore $a > b$ .
The only possible combinations of ${(a + b), (a - b)}$ are $(2^{8}, 2), (2^{7}, 2^{2}), (2^{6}, 2^{3}), (2^{5}, 2^{4})$ .
Based on above, we can make 4 set of linear equations in $a$ and $b$ which will give us $4$ solutions for $(a, b)$ .

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