Q63 / 180 · KCET 2013 · Mathematics

If the length of the subtangent at any point to the curve $x y^{n}=a$ is proportional to the abscissa, then $n$ is

  • Correct answer
    A
    any non-zero real number
  • B
    2
  • C
    $-2$
  • D
    1

Solution

Given, curve is $x y^{n}=a$ On differentiating w.r.t. $x$, we get $x \cdot n y^{n-1} \cdot \frac{d y}{d x}+y^{n}=0$ $\Rightarrow \quad y^{n-1}\left[x n \frac{d y}{d x}+y\right]=0$ $\Rightarrow \quad \frac{d y}{d x}=-\frac{y}{n x} \quad(\because y \neq 0)$ $\begin{aligned} \therefore \text { Length of subtangent } &=\frac{y}{(d y / d x)} \\ &=y \times\left(\frac{-n x}{y}\right)=-n x \end{aligned}$ Since, it is proportional to $x$. So, $n$ can be any non-zero real number.

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

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