Eliminate $x, y$ from the equations $4 (x^{2} + y^{2}) = a x + b y, 2 (x^{2} - y^{2}) = a x - b y, x y = c^{2}$ .

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Solution

From the first two equations using elimination, we get;-
$a x = 3 x^{2} + y^{2}$
$b x = x^{2} + 3 y^{2}$
$a x y = 3 x^{2} y + y^{3}$
$∵ x y = c^{2}$
$∴ a c^{2} = 3 x^{2} y + y^{3}$
Similarly
$b c^{2} = x^{3} + 3 y^{2} x$
$(a + b) c^{2} = (x + y)^{3}$
$(a - b) c^{2} = (y - x)^{3}$
${{(a + b)}^{2 / 3} - {(a - b)}^{2 / 3}} c^{2 / 3} = {(y + x)}^{2} - {(y - x)}^{2}$
$= 4 x y = c^{2}$
$∴ {(a + b)}^{2 / 3} - {(a - b)}^{2 / 3} = 4 c^{2 / 3}$

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