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If the sum of two numbers is 4 times the geometric mean then find the ratio of numbers.(a) \(\frac{8±3\sqrt{5}}{1}\)(b) \(\frac{8±3\sqrt{7}}{1}\)(c) \(\frac{6±3\sqrt{5}}{1}\)(d) \(\frac{6±3\sqrt{7}}{1}\)I got this question in a job interview.The origin of the question is Relationship Between A.M. and G.M. in chapter Sequences and Series of Mathematics – Class 11

Answer»

Right CHOICE is (b) \(\frac{8±3\sqrt{7}}{1}\)

BEST explanation: We know, G.M. of two numbers a and b is √ab.

So, a + b = 4 √ab

Squaring we GET, a^2+b^2 = 16ab

=>(a/b) + (b/a) = 16

Let x = a/b.

So, x + 1/x = 16 => x^2 – 16x + 1 = 0

=>x = \(\frac{16±\sqrt{256-4}}{2} = \frac{16±\sqrt{252}}{2} = \frac{16±6\sqrt{7}}{2} = \frac{8±3\sqrt{7}}{1}\).


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