If d_{1},d_{2}  are two GMs between two numbers x and y then

\frac{d1^{2}}{d_{2}}+\frac{d2^{2}}{d_{1}} is equal to

  • Option 1)

    x+y

  • Option 2)

    xy

  • Option 3)

    \frac{x+y}{xy}

  • Option 4)

    None of these

 

Answers (1)

As learnt in

Geometric mean of n numbers (GM) -

GM = \left ( a_{1},a_{2},a_{3},------a_{n} \right )^{1/n}

- wherein

a_{1},a_{2},a_{3},------a_{n}

are the n numbers

 

 x, d_{1}, d_{2}, y

We will find the common ratio 'r'

y=xr^{3}\Rightarrow r^{3}=\frac{y}{x} \Rightarrow r = \left ( \frac{y}{x} \right )^{\frac{1}{3}}

d_{1} = x^{\frac{2}{3}} y^{\frac{1}{3}}

d_{2} = x^{\frac{1}{3}} y^{\frac{2}{3}}

\frac{d_{1}^{2}}{d_{2}} + \frac{d_{2}^{2}}{d_{1}} = \frac{d_{1}^{3}+d_{2}^{3}}{d_{1}d_{2}}

= \frac{x^{2}y+xy^{2}}{xy} = \frac{xy(x+y)}{xy} =(x+y)


Option 1)

x+y

This solution is correct

Option 2)

xy

This solution is incorrect

Option 3)

\frac{x+y}{xy}

This solution is incorrect

Option 4)

None of these

This solution is incorrect

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