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Question

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 angle at centre. Radius joining their interface makes an angle α with vertical.
Ratio d1d2 is:

A
1+tan α1tan α
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B
1+sin α1cos α
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C
1+sin α1sin α
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D
1+cos α1cos α
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Solution

The correct option is A 1+tan α1tan α
Let the point be O of tube touching the ground. Pressure due to liquid of density d1 on the left side of O = pressure due to liquid of density d2 on the right side of O.

d1(RR sin α)=d1(RR cos α)+d2(R sin α+R cos α)
d1(1 - sinα)=d1(1cos α)+d2(sin α+cos α)
d1d1sinα=d1d1cosα+d2(sinα+cosα)
d1(cosαsin α)=d2(sinα+cosα)
d1d2=sinα+cosαcosαsinα=1+tanα1tanα

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