x and y are two numbers such that their mean proportion is 8 and third proportion is 512. What is the value of x and y?

Three numbers a,b,c are in proportion iff $$b^2 = ac$$ where $$b$$ is the mean proportion and $$c$$ is the third proportion

Mean proportion of two numbers $$x$$ and $$y$$ = 8

=> $$xy = (8)^2 = 64$$ ---------------(i)

Third proportion = 512

=> $$y^2 = x \times 512$$ -------------(ii)

Substituting value of $$x$$ from equation(i) in equation(ii), we get :

=> $$y^2 = \frac{64}{y} \times 512$$

=> $$y^3 = (2)^6 \times (2)^9 = (2)^{15}$$

=> $$y = (2)^{\frac{15}{3}} = 2^5 = 32$$

Substituting it in equation(i), => $$x = \frac{64}{32} = 2$$

=> Ans - (B)