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A particle is moving with velocity $\overrightarrow{v}=K(y\overrightarrow{i}+x\overrightarrow{j})$, where K is a constant. The general equation for its path is:
$\begin{align}
  & (A)y={{x}^{2}}+constant \\
 & (B){{y}^{2}}=x+constant \\
 & (C)xy=constant \\
 & (D){{y}^{2}}={{x}^{2}}+constant \\
\end{align}$

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Last updated date: 13th Jun 2024
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Answer
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Hint: First consider the general equation of $\overrightarrow{v}$ and compare it with the given equation of $\overrightarrow{v}$. Then we will get a value for the x and y component of $\overrightarrow{v}$. Then differentiating the x and y component of $\overrightarrow{v}$ with respect to time t. Then we get two equations. And by dividing them we will get $\dfrac{dy}{dx}$. Then by using a variable separable method and integrating we will get the final answer.

Complete answer:
Given that,
$\overrightarrow{v}=K(y\overrightarrow{i}+x\overrightarrow{j})$
$\overrightarrow{v}=Ky\overrightarrow{i}+Kx\overrightarrow{j}$ ………….(1)
Consider the general equation of $\overrightarrow{v}$,
$v={{v}_{x}}\overrightarrow{i}+{{v}_{_{y}}}\overrightarrow{j}$ ………….(2)
Comparing equation (1) and (2),
We will get like this,
${{v}_{x}}=Ky$
and
${{v}_{y}}=Kx$
Then by differentiating the x and y component of v we get,
$\dfrac{dx}{dt}=Ky$ ………….(3)
$\dfrac{dy}{dt}=Kx$ …………(4)
Now by dividing equation(4) by (3),
$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}=\dfrac{x}{y}$
Then by using variable separable method and rearranging we get,
$ydy=xdx$
Integrating on both sides we get,
$\int{ydy=\int{xdx}}$
We know that in general,
$\int{xdx=\dfrac{{{x}^{2}}}{2}}$
By using this concept it becomes,
$\dfrac{{{y}^{2}}}{2}=\dfrac{{{x}^{2}}}{2}+c$ ………………..(5)
Then multiplying equation (5) by 2 we get,
${{y}^{2}}={{x}^{2}}+2c$
where 2c is the constant of integration.
Then it becomes,
${{y}^{2}}={{x}^{2}}+$ constant.
This is the general equation for its path.

Hence, option(D) is correct.

Note:
The general equation for $\overrightarrow{v}$ is ${{v}_{x}}\overrightarrow{i}+{{v}_{y}}\overrightarrow{j}$ and compare it with the given equation of $\overrightarrow{v}$. Then we will get a value for the x and y component of $\overrightarrow{v}$. While using Variable separation method for integration, always bring x components to one side and y components to the other side. The name itself shows that. Thus we get the general equation of path .