Question

For a real variable $a > 1$, consider the points $A_{k}=\left(k a, a^{k}\right), k=1,2, \ldots n$ in the cartesian plane. If $\alpha$ and $\beta$ represent respectively the arithmetic mean of $x$ coordinates and the geometric mean of $y$ coordinates of $A_{k}$. then the locus of the point $P (\alpha, \beta)$ is

• A. $n y=\left(\frac{2 x}{n}\right)^{n^{2}+1}$
• B. $y^{2}=\left(\frac{2 x}{n+1}\right)^{n+1}$
• C. $y=\left(\frac{x^{2}}{n+1}\right)^{n}$
• D. $y+(n+1)(x-(n+1))$

Answer 1

Answer: (B)

We have
$a =\frac{a+2 a+3 a+\ldots+n a}{n} $
$=\frac{a[1+2+3+\ldots+n]}{n} $
$=\frac{a n(n+1)}{2 n}=\frac{a(n+1)}{2}$
and $\beta=\left(a \cdot a^{2} \cdot a^{3} \cdot a^{4} \ldots a^{n}\right)^{1 / n} $
$=\left(a^{1+2+3+\ldots n}\right)^{1 / n}=a^{\left(\frac{n(n+1)}{2}\right)^{1 / n}}=a^{\frac{n+1}{2}}$
Now, $\beta^{2}=a^{n+1} $
and $\left(\frac{2 \alpha}{n+1}\right)^{n+1}=\left(\frac{2 a(n+1)}{2(n+1)}\right)^{n+1}=a^{n+1} $
$\therefore \beta^{2}=\left(\frac{2 \alpha}{n+1}\right)^{n+1}$
So, locus of point $P(\alpha, \beta)$ is
$y^{2}=\left(\frac{2 x}{n+1}\right)^{n+1}$