The A.M.(Arithmetic Mean) and G.M (Geometric Mean) of two positive numbers p and q are 8 and 4 respectively, then the value of \(p-q\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\sqrt{192}\)
For two positive numbers, \(\text{AM}=\frac{p+q}{2}\) and \(\text{GM}=\sqrt{pq}\). The AM gives the sum and the GM gives the product, and the difference follows from an identity: \((p-q)^{2}=(p+q)^{2}-4pq\).
The AM is 8, so the sum is \(p+q=16\)
The GM is 4, so the product is \(pq=16\)
Square of the difference: \((p-q)^{2}=16^{2}-4\times16=256-64=192\)
So \(p-q=\sqrt{192}\), taking \(p\) as the larger number (\(\sqrt{192}=8\sqrt{3}\))