InterviewSolution
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InterviewSolution
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The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = `(m+sqrt(m^2-n^2)):(m-sqrt(m^2-n^2))`. |
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Answer» Here, we are given,`((a+b)/2)/sqrt(ab) = m/n` `=>(a+b)/(2sqrt(ab)) = m/n->(1)` Squaring both sides, `=>((a+b)^2)/(4ab) = m^2/n^2` `=>((a+b)^2-4ab)/(4ab) = (m^2-n^2)/n^2` `=>((a-b)^2)/(4ab) = (m^2-n^2)/n^2` `=>(a-b)/(2sqrt(ab)) = sqrt(m^2-n^2)/n->(2)` Adding (1) and (2), `(2a)/(2sqrt(ab)) = (m+sqrt(m^2-n^2))/n ->(3)` Subtracting (2) from (1), `(2b)/(2sqrt(ab)) = (m-sqrt(m^2-n^2))/n ->(4)` Dividing (3) by (4), `a/b = (m+sqrt(m^2-n^2))/ (m-sqrt(m^2-n^2))` So, `a:b = (m+sqrt(m^2-n^2)) : (m-sqrt(m^2-n^2))` |
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