In a triangle the sum of two sides is x and the product of the same two sides is y. If x2−c2=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 B3y2c(x+c) a+b=xab=yx2−c2=y
Using cosine rule, cosC=a2+b2−c22ab⇒cosC=(a+b)2−2ab−c22y⇒cosC=x2−2y−c22y=−12⇒C=2π3
Now, csinC=2R⇒R=c√3r=(s−c)tanC2⇒r=√3(a+b−c2)=√3(x−c)2
Now, rR=3(x−c)2c=3y2c(x+c)