Question

A particle is moving with a velocity $\overrightarrow{v}=K(y\hat{i}+x\hat{j}),$ where $K$ is a constant. The general equation for its path is:

• A. $y^2=x^2+\text{constant}$
• B. $y^2=x+\text{constant}$
• C. $xy=\text{constant}$
• D. $y=x^2+\text{constant}$

Answer 1

The correct option is A $y^2=x^2+\text{constant}$

From given equation,
$\overrightarrow{v}=K(y\hat{i}+x\hat{j})$

$v_x=\frac{dx}{dt}=Ky\text{and}{v}_y=\frac{dy}{dt}=Kx$

$\Rightarrow \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{x}{y}$

$\Rightarrow \frac{dy}{dx}=\frac{x}{y}$

$\Rightarrow ydy=xdx$

Integrating both sides,

$y^2=x^2+c$

Hence, option $\left(C\right)$ is correct.