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Question: Answered & Verified by Expert
If the arithmetic mean of two numbers be $A$ and geometric mean be $G$, then the numbers will be
MathematicsSequences and SeriesJEE Main
Options:
  • A $A \pm\left(A^2-G^2\right)$
  • B $\sqrt{A} \pm \sqrt{A^2-G^2}$
  • C $A \pm \sqrt{(A+G)(A-G)}$
  • D $\frac{A \pm \sqrt{(A+G)(A-G)}}{2}$
Solution:
1914 Upvotes Verified Answer
The correct answer is: $A \pm \sqrt{(A+G)(A-G)}$
A.M. $=\frac{a+b}{2}=A$ and G.M. $=\sqrt{a b}=G$
On solving $a$ and $b$ are given by the values
$A \pm \sqrt{(A+G)(A-G)}$
Trick : Let the numbers be 1,9 . Then $A=5$ and $G=3$. Now put these values in options.
Here (3) $\Rightarrow 5 \pm \sqrt{8 \times 2}$ i.e. 9 and 1 .

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