Sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+22 ):(3−22 ).

Question

Sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+22​ ):(3−22​ ).

Filo · Accepted Answer

Let the numbers be a and b.Geometric mean of two numbers a and b is ab​.According to the question,a+b=6ab​2ab​a+b​=13​Apply componendo and dividendo.a+b−2ab​a+b+2ab​​=3−13+1​(a​)2+(b​)2−2(a​×b​)(a​)2+(b​)2+2(a​×b​)​=24​(a​−b​a​+b​​)2=12​a​−b​a​+b​​=12​​Apply componendo and dividendo again.(a​−b​)−(a​−b​)(a​+b​)+(a​−b​)​=2​−12​+1​a​−a​+b​+b​a​+a​+b​−b​​=2​−12​+1​2b​2a​​=2​−12​+1​=2​−12​+1​ba​​=2​−12​+1​Square both the sides.(ba​​)2=(2​−12​+1​)2ba​=(2​−12​+1​)ba​=3−22​3+22​​Thus, a and b are in the ratio of 3+22​:3−22​.

Question Text	Sum of two numbers is $6$ times their geometric mean, show that numbers are in the ratio $(3 + 22) : (3 - 22)$ .
Answer Type	Text solution:1
Upvotes	150

Sum of two numbers is $6$ times their geometric mean, show that numbers are in the ratio $(3 + 22) : (3 - 22)$ .

Text solutionVerified

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Sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio (3+22​ ):(3−22​ ).

Text solutionVerified

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Sum of two numbers is $6$ times their geometric mean, show that numbers are in the ratio $(3 + 22) : (3 - 22)$ .