There is a circular tube in a vertical plane. Two liquids which do not mix and of densities d₁ and d₂ are filled in the tube. Each liquid subtends 90^o angle at centre. Radius joining their interface makes an angle α with vertical. ratio d₁/d₂ is

$fraction numerator 1 plus sin space straight alpha over denominator 1 minus sin space straight alpha end fraction$
$fraction numerator 1 plus space cosα over denominator 1 minus cosα end fraction$
$fraction numerator 1 plus space tan space straight alpha over denominator 1 minus tan space straight alpha end fraction$
$fraction numerator 1 plus cos space straight alpha over denominator 1 minus cos space straight alpha end fraction$

Solution

fraction numerator 1 plus space tan space straight alpha over denominator 1 minus tan space straight alpha end fraction

straight P subscript straight A space equals space straight P subscript straight B straight P subscript 0 space plus space straight d subscript 1 straight g end subscript straight R space left parenthesis Cos space straight alpha space minus sin space straight alpha right parenthesis space equals straight P subscript 0 space plus straight d subscript 2 straight R space left parenthesis Cos space straight alpha plus sin space straight alpha right parenthesis rightwards double arrow space straight d subscript 1 over straight d subscript 2 space equals space fraction numerator cos space straight alpha space plus sin space straight alpha over denominator cos space straight alpha minus sin space straight alpha end fraction space equals space fraction numerator 1 plus space tan space straight alpha over denominator 1 minus space tanα end fraction

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