The mean proportion between x and y is 6. The third proportional to x and y is 48. Then x : y equals :

1. 1 : 3

2. 1 : 4

3. 1 : 6

4. 1 : 9

Given,

Mean proportion between x and y is 6.

⇒ x : 6 = 6 : y

⇒ $\dfrac{x}{6} = \dfrac{6}{y}$

⇒ y = $\dfrac{36}{x}$ ……(1)

Given,

The third proportional to x and y is 48.

⇒ x : y = y : 48

⇒ $\dfrac{x}{y} = \dfrac{y}{48}$

⇒ y² = 48x …….(2)

Substitute value of y from equation 1 in equation 2

$\Rightarrow 48x = \Big(\dfrac{36}{x}\Big)^2 \\[1em] \Rightarrow 48x = \Big(\dfrac{1296}{x^2}\Big) \\[1em] \Rightarrow 48x \times x^2 = 1296 \\[1em] \Rightarrow 48x^3 = 1296 \\[1em] \Rightarrow x^3 = \dfrac{1296}{48} \\[1em] \Rightarrow x^3 = 27 \\[1em] \Rightarrow x = \sqrt[3]{27} \\[1em] \Rightarrow x = 3.$

Substituting value of x in equation 1, we get :

⇒ y = $\dfrac{36}{3}$

⇒ y = 12

⇒ x : y = 3 : 12

⇒ x : y = 1 : 4.

Hence, option 2 is the correct option.