Question

For a real variable $a>1$, consider the points $A_k=\left(ka,a^k\right),k=1,2,\dots n$ 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. $ny={\left(\frac{2x}{n}\right)}^{n^2+1}$
• B. $y^2={\left(\frac{2x}{n+1}\right)}^{n+1}$
• C. $y={\left(\frac{x^2}{n+1}\right)}^n$
• D. $y+(n+1)(x-(n+1))$

Answer 1

Answer: (B)

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