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Let a smooth curve $\mathrm{y=f(x)}$ be such that the slope of the tangent at any point $\mathrm{(x, y)}$ on it is directly proportional to $\mathrm{\left(\frac{-y}{x}\right)}$ . If the curve passes through the points $\mathrm{(1,2) \: and\: (8,1)}$ , then $\mathrm{\left|y\left(\frac{1}{8}\right)\right|}$ is equal to
Option: 1
$2\: log_{e}\: 2$

Option: 2
$4$

Option: 3
$1$

Option: 4
$4\: log_{e}\: 2$

Answers (1)

best_answer

$\mathrm{\frac{d y}{d x} =k\left(-\frac{y}{x}\right)} \\$

$\mathrm{\Rightarrow \frac{d y}{y} =-k \frac{d x}{x} }\\$

$\mathrm{\Rightarrow \ln y =-k \ln x+\ln C} \\$

$\mathrm{\Rightarrow y =c x^{-k}}$

$\mathrm{(1,2) \Rightarrow 2=c} \\$

$\mathrm{(8,1) \Rightarrow 1=2 \cdot(8)^{-k}} \\$

$\mathrm{\Rightarrow 8^{k}=2} \\$

$\mathrm{\Rightarrow k=\frac{1}{3}} \\$

$\mathrm{\therefore y=2 x^{-\frac{1}{3}}} \\$

$\mathrm{\left.\therefore \mid y\left(\frac{1}{8}\right)\right)=2 \cdot\left(\frac{1}{8}\right)^{-\frac{1}{3}}=2 \cdot 2=4}$

Hence correct option is 2

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