Let ‘a’ be a fixed positive number and ‘n’ be an arbitrary constant. For the curve y=xnan−1 if the length of the subnormal at any point α, β is proportional to α2, then n = ____

y=xnan−1dydx=n.xn−1an−1Length of subnormal = y.dydx=xnan−1.n.xn−1an−1=n.x2n−1a2n−2Length of subnormal at (α,β)=n.αa2n−2Length of subnormal is proportional to a2 ⇒2n−1=2⇒n=32.

Let ‘a’ be a fixed positive number and ‘n’ be an arbitrary constant. For the curve y=xnan−1 if the length of the subnormal at any point α, β is proportional to α2, then n = ____

y=xnan−1dydx=n.xn−1an−1Length of subnormal = y.dydx=xnan−1.n.xn−1an−1=n.x2n−1a2n−2Length of subnormal at (α,β)=n.αa2n−2Length of subnormal is proportional to a2 ⇒2n−1=2⇒n=32.

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Let ‘a’ be a fixed positive number and ‘n’ be an arbitrary constant. For the curve $y = \frac{x^{n}}{a^{n - 1}}$ if the length of the subnormal at any point $(α, β)$ is proportional to $α^{2}$ , then n = ____

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$\frac{3}{2}$

answer is A.

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Detailed Solution

$\begin{array}{l} y = \frac{x^{n}}{a^{n - 1}} \\ \frac{d y}{d x} = \frac{n . x^{n - 1}}{a^{n - 1}} \end{array}$

Length of subnormal = $y . \frac{d y}{d x} = \frac{x^{n}}{a^{n - 1}} . \frac{n . x^{n - 1}}{a^{n - 1}} = \frac{n . x^{2 n - 1}}{a^{2 n - 2}}$
Length of subnormal at $(α, β) = \frac{n . α}{a^{2 n - 2}}$
Length of subnormal is proportional to $a^{2}$
$\Rightarrow 2 n - 1 = 2 \Rightarrow n = \frac{3}{2}$ .