Q67 / 160 · WBJEE 2010 · Mathematics
The value of $\mathrm{n}$ for which $\frac{\mathrm{x}^{\mathrm{n}+1}+\mathrm{y}^{\mathrm{n}+1}}{\mathrm{x}^{\mathrm{n}}+\mathrm{y}^{\mathrm{n}}}$ is the geometric mean of $\mathrm{x}$ and $\mathrm{y}$ is
- Correct answerA$\mathrm{n}=-\frac{1}{2}$
- B$\mathrm{n}=\frac{1}{2}$
- C$\mathrm{n}=1$
- D$\mathrm{n}=-1$
Solution
Hints: $\frac{x^{n+1}+y^{n+1}}{x^n+y^n}=\sqrt{x y} \Rightarrow x^{n+1}+y^{n+1}=\sqrt{x y}\left(x^n+y^n\right)$
$$
\mathrm{x}^{\mathrm{n}+\frac{1}{2}}\left(\mathrm{x}^{\frac{1}{2}}-\mathrm{y}^{\frac{1}{2}}\right)=\mathrm{y}^{\mathrm{n}+\frac{1}{2}}\left(\mathrm{x}^{\frac{1}{2}}-\mathrm{y}^{\frac{1}{2}}\right),\left(\frac{\mathrm{x}}{\mathrm{y}}\right)^{\mathrm{n}+\frac{1}{2}}=1 \quad \mathrm{n}=-\frac{1}{2}
$$
Hence, the correct answer is option: 'A'.