Find $a^{2}$ and $b^{2}$
i) $16 b^{2} + 9 a^{2} = a^{2} b^{2}$
ii) $36 b^{2} + 4 a^{2} = a^{2} b^{2}$

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Solution

Equating both equations we get,
$36 b^{2} + 4 a^{2} = 16 b^{2} + 9 a^{2}$
$20 b^{2} = 5 a^{2}$
$4 b^{2} = a^{2}$
$a = \pm 2 b$
Equating this in either equation
$16 b^{2} + 36 b^{2} = 4 b^{4}$
$52 b^{2} = 4 b^{4}$
$⟹ 13 = b^{2} = \pm \sqrt{13}$
$⟹ a^{2} = 52 = \pm \sqrt{52}$

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