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If a geometric mean of two non-negative numbers is equal to their harmonic mean, then which of the following is necessarily true?I. One of the numbers is zero.II. Both the numbers are equal.III. One of the numbers is one.
- a)I and III only
- b)Either I or III
- c)III only
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
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If a geometric mean of two non-negative numbers is equal to their harm...
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Geometric Mean and Harmonic Mean
Before we start solving the problem, let's first understand what is meant by geometric mean and harmonic mean.
Geometric Mean: The geometric mean of two non-negative numbers a and b is the square root of their product, i.e., GM = √(ab).
Harmonic Mean: The harmonic mean of two non-negative numbers a and b is the reciprocal of their arithmetic mean, i.e., HM = 2/(1/a + 1/b) = 2ab/(a+b).
Solving the Problem
Now, let's try to solve the given problem.
GM = HM
√(ab) = 2ab/(a+b)
Squaring both sides, we get:
ab = 4a^2b^2/(a+b)^2
Multiplying both sides by (a+b)^2, we get:
ab(a+b)^2 = 4a^2b^2
Simplifying, we get:
a^2 + 2ab + b^2 = 4ab
a^2 - 2ab + b^2 = 0
(a-b)^2 = 0
a = b
Therefore, both the numbers are equal.
Conclusion
Hence, we can conclude that if the geometric mean of two non-negative numbers is equal to their harmonic mean, then both the numbers are equal. None of the given options (I, II, and III) is necessarily true.
Before we start solving the problem, let's first understand what is meant by geometric mean and harmonic mean.
Geometric Mean: The geometric mean of two non-negative numbers a and b is the square root of their product, i.e., GM = √(ab).
Harmonic Mean: The harmonic mean of two non-negative numbers a and b is the reciprocal of their arithmetic mean, i.e., HM = 2/(1/a + 1/b) = 2ab/(a+b).
Solving the Problem
Now, let's try to solve the given problem.
GM = HM
√(ab) = 2ab/(a+b)
Squaring both sides, we get:
ab = 4a^2b^2/(a+b)^2
Multiplying both sides by (a+b)^2, we get:
ab(a+b)^2 = 4a^2b^2
Simplifying, we get:
a^2 + 2ab + b^2 = 4ab
a^2 - 2ab + b^2 = 0
(a-b)^2 = 0
a = b
Therefore, both the numbers are equal.
Conclusion
Hence, we can conclude that if the geometric mean of two non-negative numbers is equal to their harmonic mean, then both the numbers are equal. None of the given options (I, II, and III) is necessarily true.
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If a geometric mean of two non-negative numbers is equal to their harmonic mean, then which of the following is necessarily true?I. One of the numbers is zero.II. Both the numbers are equal.III. One of the numbers is one.a)I and III onlyb)Either I or IIIc)III onlyd)None of theseCorrect answer is option 'D'. Can you explain this answer?
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