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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+sinα1sinα
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B
1+cosα1cosα
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C
1+tanα1tanα
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D
1+sinα1cosα
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Solution

The correct option is C 1+tanα1tanα
As we know pressure on line AB P=P0+ρghasXOD=90&DOE=α
XOC also α So by trignometry y1=Rsinαcosα=y2R
P=P0+d2gRsinα+d2gRcosαP=P0+d2gR{sinα+cosα}(i)
y3=RcosαRsinαP=P0+d1g{RcosαRsinα}P=P0+d1gR{cosαsinα}
From eq (1) & (2)
P0+d2gR{sinα+cosα}=P0+d1gR{cosαsinα}
d1d2=sinα+cosαcosαsinα
d1d2=1+tanα1tanα

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