Q73 / 180 · COMEDK 2013 · Mathematics

For the curve $y^{n}=a^{n-1} x$ if the subnormal at any point is a constant, then $n$ is equal to

  • A
    $1$
  • Correct answer
    B
    $2$
  • C
    $-2$
  • D
    $-1$

Solution

We have, $y^{n}=a^{n-1} x$ $\Rightarrow \quad n \cdot y^{n-1} \frac{d y}{d x}=a^{n-1}$ $\Rightarrow \quad y^{n-1} \frac{d y}{d x}=\frac{1}{n} a^{n-1}$ $\Rightarrow \quad \frac{d y}{d x}=\frac{1}{n} a^{n-1} y^{1-n}$ $\therefore \quad$ Length of the subnormal $=\left|y \frac{d y}{d x}\right|$ $\quad=\left|y \times \frac{1}{n} a^{n-1} y^{1-n}\right|$ $\quad=\left|\frac{1}{n} a^{n-1} y^{2-n}\right|$ $\therefore \quad 2-n=0$ $\Rightarrow \quad n=2$

Hence, the correct answer is option: 'B'.

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