Find mean proportional between \(\begin{array}{l} \left( {15 + \sqrt {200} } \right){\rm{\ and\ }}\left( {27 - \sqrt {648} } \right)\\ \end{array}\)
Solution
Correct Answer: Option B
The mean proportion between ‘a’ and ‘b’ is √ab, if mean proportion is ‘x’, then,
⇒ a : x = x : b
⇒ a/x = x/b
⇒ x2 = ab
Or, √ab = x
Now, let the required mean proportion be ‘m’, then
\(\left( {15 + \sqrt {200} } \right){\rm{}}:{\rm{m}} = {\rm{m}}:\left( {27 - \sqrt {648} } \right)\)
According to the rule,
The required mean proportional,
\(\begin{array}{l} \Rightarrow m = \sqrt {\left( {15 + \sqrt {200} } \right) \times \left( {27 - \sqrt {648} } \right)} \\ \Rightarrow m = \sqrt {\left( {15 \times 27 - 15 \times \sqrt {648} } \right) + \left( {27 \times \sqrt {200} - \sqrt {200} \times \sqrt {648} } \right)} \\ \Rightarrow m = \sqrt {405 - \left( {15 \times 18\sqrt 2 } \right) + \left( {27 \times 10\sqrt 2 } \right) - \left( {10\sqrt 2 \times 18\sqrt 2 } \right)} \\ \Rightarrow m = \sqrt {405 - 270\sqrt 2 + 270\sqrt 2 - 180 \times 2} \\ \Rightarrow m = \sqrt {405 - 360} = \sqrt {45} \\ \Rightarrow m = 3\sqrt 5 \end{array}\)
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