If the length of the subtangent at any point | vedclass

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(A) Given,the curve is $x y^{n} = a$. On differentiating with respect to $x$,we get: $x \cdot n y^{n-1} \cdot \frac{dy}{dx} + y^{n} = 0$ $\Rightarrow

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any non-zero real number

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(A) Given,the curve is $x y^{n} = a$. On differentiating with respect to $x$,we get: $x \cdot n y^{n-1} \cdot \frac{dy}{dx} + y^{n} = 0$ $\Rightarrow y^{n-1} [x n \frac{dy}{dx} + y] = 0$ Since $y 
eq 0$,we have $x n \frac{dy}{dx} + y = 0$,which implies $\frac{dy}{dx} = -\frac{y}{nx}$. The length of the subtangent is given by $|\frac{y}{dy/dx}|$. Substituting the value of $\frac{dy}{dx}$: Length of subtangent $= |\frac{y}{-y/nx}| = |nx| = |n| |x|$. Since the length of the subtangent is proportional to the abscissa $x$,the value $|n|$ must be a constant. Thus,$n$ can be any non-zero real number.

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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