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If the length of the subtangent at any point to the curve $x y^{n}=a$ is proportional to the abscissa, then $n$ is
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The correct answer is:
any non-zero real number
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.
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.
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