Question

A point is in motion along a hyperbola $y=\frac{10}{x}$ so that its abscissa $x$ increases uniformly at a rate of $1$ unit per second. Then, the rate of change of its ordinate, when the point passes through $(5,2)$

• A. increases at the rate of $\frac{1}{2}$ unit per second
• B. decreases at the rate of $\frac{1}{2}$ unit per second
• C. decreases at the rate of $\frac{2}{5}$ unit per second
• D. increases at the rate of $\frac{2}{5}$ unit per second

Answer 1

The correct option is C decreases at the rate of $\frac{2}{5}$ unit per second

Let $x$ and $y$ be the functions of parameter time, $t$
$\Rightarrow \frac{dx}{dt}=1$

$y=\frac{10}{x}$
$\frac{dy}{dt}=\frac{-10}{x^2}\cdot \frac{dx}{dt}$
$\Rightarrow \frac{dy}{dt}=\frac{-10}{x^2}\Rightarrow \frac{dy}{dt}{|}_{x=5}=-\frac{10}{25}=-\frac{2}{5}$
$\Rightarrow$ ordinate deceases at the rate of $\frac{2}{5}$ unit per second