A particle moves along the curve

Question:

A particle moves along the curve $y=x^{2}+2 x .$ At what point(s) on the curve are the $x$ and $y$ coordinates of the particle changing at the same rate?

Solution:

Here,

$y=x^{2}+2 x$

$\Rightarrow \frac{d y}{d t}=(2 x+2) \frac{d x}{d t}$

$\Rightarrow 2 x+2=1$             $\left[\because \frac{d y}{d t}=\frac{d x}{d t}\right]$

$\Rightarrow 2 x=-1$

$\Rightarrow x=\frac{-1}{2}$

Substituting $x=\frac{-1}{2}$ in $y=x^{2}+2 x$, we get

$y=\frac{-3}{4}$

Hence, the coordinates of the point are $\left(\frac{-1}{2}, \frac{-3}{4}\right)$.

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