For a real variable a 1,consider the points | vedclass

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(B) The $x$-coordinates of the points $A_k$ are $a, 2a, 3a, \ldots, na$. The arithmetic mean $\alpha$ is given by:$\alpha = \frac{a + 2a + 3a + \ldots

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$y^2 = \left(\frac{2 x}{n+1}\right)^{n+1}$

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(B) The $x$-coordinates of the points $A_k$ are $a, 2a, 3a, \ldots, na$. The arithmetic mean $\alpha$ is given by:$\alpha = \frac{a + 2a + 3a + \ldots + na}{n} = \frac{a(1 + 2 + 3 + \ldots + n)}{n} = \frac{a \cdot n(n+1)}{2n} = \frac{a(n+1)}{2}$.Thus,$a = \frac{2\alpha}{n+1}$.The $y$-coordinates of the points $A_k$ are $a^1, a^2, a^3, \ldots, a^n$. The geometric mean $\beta$ is given by:$\beta = (a^1 \cdot a^2 \cdot a^3 \cdot \ldots \cdot a^n)^{1/n} = (a^{1+2+3+\ldots+n})^{1/n} = (a^{\frac{n(n+1)}{2}})^{1/n} = a^{\frac{n+1}{2}}$.Squaring both sides,we get $\beta^2 = a^{n+1}$.Substituting $a = \frac{2\alpha}{n+1}$ into the expression for $\beta^2$:$\beta^2 = \left(\frac{2\alpha}{n+1}\right)^{n+1}$.Replacing $(\alpha, \beta)$ with $(x, y)$,the locus is $y^2 = \left(\frac{2x}{n+1}\right)^{n+1}$.

For a real variable $a > 1$,consider the points $A_k = (k a, a^k)$,$k = 1, 2, \ldots, n$ 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

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