MathematicsThe sum of two numbers is 6 times their geometric mean, then the numbers are in the ratio (3+22):(3-22). State true or false.

The sum of two numbers is 6 times their geometric mean, then the numbers are in the ratio (3+22):(3-22). State true or false.


  1. A
    True
  2. B
    False 

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    Solution:

    The above statement is true.
    First, let  two numbers be x and y
    Geometric Mean = xy
    As per the question,
    x + y = 6xy…………(i)
    + y2= 36 (xy)
    By subtracting 4xy from both the side we get,
    + y2 - 4xy = 36xy - 4xy
    x - y = 32 xy
    x - y = 42 xy……..(ii)
    By adding both equation (i & ii)  we get,
    x + y + x - y  = 6xy + 42 xy
    2x = (6 + 42) xy
    x = (3 + 22) xy
    Now lets substitute the value of x in our equation (i),
    x + y = 6xy
    (3 + 22) xy + y =  6xy
    y = 6xy - (3 + 22) xy
    y = (3 - 22) xy
    Then, xy = (3 + 22) xy(3 - 22) xy = (3 + 22) (3 - 22)
    ∴ The required Ratio = (3+22):(3-22).
     
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