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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 .

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