To determine the ratio \(a : b\) where \(a\) and \(b\) are distinct positive real numbers, given that their arithmetic mean is twice their geometric mean, we first define these means:
The arithmetic mean (AM) of \(a\) and \(b\) is \(\text{AM} = \frac{a + b}{2}\).
The geometric mean (GM) of \(a\) and \(b\) is \(\text{GM} = \sqrt{ab}\).
The problem states that \(\text{AM} = 2 \times \text{GM}\):
\(\frac{a + b}{2} = 2\sqrt{ab}\)
Multiplying by 2 yields:
\(a + b = 4\sqrt{ab}\)
Squaring both sides to remove the radical:
\((a + b)^2 = 16ab\)
Expanding the left side gives:
\(a^2 + 2ab + b^2 = 16ab\)
Rearranging the terms results in:
\(a^2 + b^2 = 14ab\)
To find the ratio \(\frac{a}{b}\), let \(\frac{a}{b} = k\), which implies \(a = kb\). Substituting \(a = kb\) into the equation:
\((kb)^2 + b^2 = 14(kb)(b)\)
Simplifying the equation:
\(k^2b^2 + b^2 = 14kb^2\)
Factoring out \(b^2\):
\((k^2 + 1)b^2 = 14kb^2\)
Dividing by \(b^2\) (since \(b eq 0\)):
\(k^2 + 1 = 14k\)
Rearranging into a quadratic equation:
\(k^2 - 14k + 1 = 0\)
Using the quadratic formula \(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -14\), and \(c = 1\):
\(k = \frac{14 \pm \sqrt{(-14)^2 - 4(1)(1)}}{2}\)
\(k = \frac{14 \pm \sqrt{196 - 4}}{2}\)
\(k = \frac{14 \pm \sqrt{192}}{2}\)
\(k = \frac{14 \pm 8\sqrt{3}}{2}\)
\(k = 7 \pm 4\sqrt{3}\)
Since \(a\) and \(b\) are positive, we consider positive ratios. If we assume \(a>b\), then \(k>1\). The value \(k = 7 + 4\sqrt{3}\) is positive and greater than 1.
Therefore, the ratio \(a : b = k : 1 = 7 + 4\sqrt{3} : 1\).
To match this ratio with potential options, we can check specific values. Given \(a = 2 + \sqrt{3}\) and \(b = 2 - \sqrt{3}\), we calculate their ratio:
Let \(k = \frac{a}{b} = \frac{2+\sqrt{3}}{2-\sqrt{3}}\). Rationalizing the denominator by multiplying by \(2 + \sqrt{3}\):
\(k = \frac{(2+\sqrt{3})(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}\)
\(k = \frac{(2+\sqrt{3})^{2}}{2^2-(\sqrt{3})^{2}}\)
\(k = \frac{4+4\sqrt{3}+3}{4-3} = \frac{7+4\sqrt{3}}{1} = 7+4\sqrt{3}\)
This confirms that the ratio \(a : b\) is equivalent to \(2+\sqrt{3} : 2-\sqrt{3}\).