Question

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Answer 1

Question :-

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 ?

SoluTion :-

$\boxed {\sf {In-radius =\frac{Area}{Semi-Perimeter} }}\ Or \ \boxed {\sf {\frac{\Delta}{s} }}\\\\\\\\\boxed {\sf {Circum-radius = \frac{abc}{4 \Delta} }}\\\\\\\\\sf {\rightarrowtail \ Ratio = \frac{4 \Delta^{2}}{abcs} }\\\\\\\\\sf {\rightarrowtail \frac{8 \Delta^{2}}{yc(x+c)} }\\\\\\\\\\\sf {\Delta = \frac{1}{2}ab.sinC }\\\\\\\\\sf {\Rightarrow CosC=\frac{a^{2}+b^{2}-c^{2}}{2ab} }\\\\\\\sf {\Rightarrow CosC=\frac{x^{2}-2y-c^{2}}{2y} }\\\\\\\sf {\Rightarrow CosC=\frac{-1}{2} }$

⇒ C = 120°

$\sf {\therefore \ SinC=\frac{\sqrt{3} }{2} }\\\\\\\sf {\Delta = \frac{1}{2}y \ \times \ \frac{\sqrt{3}}{2} }\\\\\\\sf {\Delta =\frac{\sqrt{3} }{4}y }$

Substituting,

$\boxed {\sf {Ratio = \frac{3y}{2c(x+c)} }}$

Option B is Correct.

Answer 2

Answer:

the above answer is correct

Step-by-step explanation:

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