Find the mean proportional between (4+√5) and (12-2√20).

Correct option is B

Given:
Find the mean proportional between (4 + √5) and (12 − 2√20).
Formula Used:
Mean Proportional between two numbers a and b = $$\sqrt{a \times b}$$​​
Solution:
Let:
​$$a = (4 + \sqrt{5}), \quad b = (12 - 2\sqrt{20})$$​​
​$$\sqrt {a \times b} = \sqrt {(4 + \sqrt{5})(12 - 2\sqrt{20})}$$ 
​$$\sqrt{20} = 2\sqrt{5}$$​​
​$$b = 12 - 2 \times 2\sqrt{5} = 12 - 4\sqrt{5}$$ 
$$\text{Mean Proportional} = \sqrt {(4 + \sqrt{5})(12 - 4\sqrt{5})}$$​​
Mean Proportional = $$\sqrt {4 \times 12 - 4 \times 4\sqrt{5} + \sqrt{5} \times 12 - \sqrt{5} \times 4\sqrt{5}}$$​​
​$$\text{Mean Proportional}= \sqrt {48 - 16\sqrt{5} + 12\sqrt{5} - 4 \times 5}\\\text{Mean Proportional}= \sqrt {48 - 4\sqrt{5} - 20}\\\text{Mean Proportional}= \sqrt {28 - 4\sqrt{5}}\\\text{Mean Proportional} = 2\sqrt {7-\sqrt 5}$$​​