TS EAMCET · Maths · Sequences and Series
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 & Solution
Correct Answer
(B) \(y^2=\left(\frac{2 x}{n+1}\right)^{n+1}\)
Step-by-step Solution
Detailed explanation
We have, \(\begin{aligned} \alpha & =\frac{a+2 a+3 a+\ldots+n a}{n} \\ & =\frac{a[1+2+3+\ldots+n]}{n} \\ & =\frac{a n(n+1)}{2 n}=\frac{a(n+1)}{2}\end{aligned}\)…
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