Q105 / 155 · WBJEE 2019 · Mathematics

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
  • Correct answer
    C
    decreases at the rate of $\frac{2}{5}$ unit per second
  • D
    increases at the rate of $\frac{2}{5}$ unit per second

Solution

We have, $$ \begin{array}{r} \frac{d x}{d t}=\operatorname{l and} y=\frac{10}{x} \\ \text { Now, } y=\frac{10}{x} \end{array} $$ On differentiating both the sides w.r.t. $t$, we get $$ \begin{aligned} \Rightarrow \quad \frac{d y}{d t} &=\frac{-10}{x^{2}} \times \frac{d x}{d t} \\ &=\frac{-10}{(5)^{2}} \times 1 \end{aligned} $$ $$ =\frac{-10}{25}=\frac{-2}{5} \quad\left[x=5, \frac{d x}{d t}=1\right] $$ $\therefore y$ decreases at the rate of $\frac{2}{5}$ unit per second.

Hence, the correct answer is option: 'C'.

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