Question

The coordinates of the point $A_n$ are $(na,a^n)$. If $\alpha$ denotes the arithmetic mean of $x$ coordinates of the points $A_1,A_2...A_n$ and $\beta$ denotes the geometric mean of the ordinates of these points, then locus of the point $P(\alpha ,\beta )$ is

Note: Consider $a$ as a variable.

• A. $x^n=n^n{y}^2$
• B. $x^{n+1}={\left(\frac{n+1}{2}\right)}^{n+1}{y}^2$
• C. $x^n=n^{n}y$
• D. None of these

Answer 1

The correct option is B $x^{n+1}={\left(\frac{n+1}{2}\right)}^{n+1}{y}^2$

$\alpha =\frac{a+2a+3a+...+na}{n}=\frac{n(n+1)a}{2n}=\frac{(n+1)a}{2}$

$\beta =\sqrt[n]{n{a}^1{a}^2{a}^3...a^n}=a^{\frac{n(n+1)}{2n}}=a^{\frac{n+1}{2}}$
Thus,

$x=\frac{(n+1)a}{2}$
$y=a^{\frac{n+1}{2}}$
$\therefore y={\left(\frac{2x}{n+1}\right)}^{\frac{n+1}{2}}$
Squaring both sides, we have $y^2={\left(\frac{2x}{n+1}\right)}^{n+1}$
$\Rightarrow x^{n+1}={\left(\frac{n+1}{2}\right)}^{n+1}\times y^2$
Hence, option $B$ is correct.