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Question

In a triangle the sum of two sides is x and the product of the same two sides is y. If x2c2=y, where c is the third side of the triangle, then the ratio of the in-radius to the circum-radius of the triangle is

A
3y2x(x+c)
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B
3y2c(x+c)
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C
3y4x(x+c)
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D
3y4c(x+c)
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Solution

The correct option is B 3y2c(x+c)
a+b=xab=yx2c2=y
Using cosine rule,
cosC=a2+b2c22abcosC=(a+b)22abc22ycosC=x22yc22y=12C=2π3
Now,
csinC=2RR=c3r=(sc)tanC2r=3(a+bc2)=3(xc)2
Now,
rR=3(xc)2c=3y2c(x+c)

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