Question

A particle is moving with velocity $\overrightarrow{v}=k(y\hat{i}+x\hat{j})$ where $k$ is constant. The general equation of the path $(c=constant)$ is

• A. $y=x^2+constant$
• B. $y^2=x+constant$
• C. $xy=constant$
• D. $y^2=x^2+constant$

Answer 1

The correct option is D $y^2=x^2+constant$

$\overrightarrow{v}=k(y\hat{i}+x\hat{j})$
$\Rightarrow v_x=ky$
Or $\frac{dx}{dt}=ky.....\left(i\right)$
and $v_y=kx$
or $\frac{dy}{dt}=kx.....\left(ii\right)$
from (i) and (ii)
$\frac{dy}{dx}=\frac{x}{y}$
$ydy=xdx$
$\Rightarrow y^2=x^2+constant$.