KCET · Maths · Application of Derivatives
If the length of the subtangent at any point to the curve \(x y^{n}=a\) is proportional to the abscissa, then \(n\) is
- A any non-zero real number
- B 2
- C \(-2\)
- D 1
Answer & Solution
Correct Answer
(A) any non-zero real number
Step-by-step Solution
Detailed explanation
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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