# A particle is moving with velocity $\dpi{100} \vec{v}=K(y\hat{i}+x\hat{j}),where\; K$ is a constant. The general equation for its path is Option 1) $y^{2}=x^{2}+constant$ Option 2) $y=x^{2}+constant$ Option 3) $y^{2}=x+constant$ Option 4) $xy=constant$

As we learnt in

Uniform velocity,Non uniform Velocity -

Uniform velocity

If equal displacement are covered in equal intervals of time.

Non uniform velocity

If equal displacement are covered in unequal intervals of time.

- wherein

More to know :

For uniform motion along a straight line, the average and instantaneous velocities have the same values.

$\vec{V}=K \left ( yi+xj \right ) - \left ( i \right )$

$\vec{V}=kyi+kxj \Rightarrow \vec{V}= \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} -\left ( ii \right )$

equation (i) and (ii)

$\frac{dx}{dt}=ky$

$\frac{dy}{dt}=kx$

$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}$

$\frac{dy}{dx}=\frac{kx}{ky}=\frac{x}{y} -\left ( iii \right )$

integrating both sides of above equation

$\int ydy= \int xdx\Rightarrow y ^2=x^2+constant$

Option 1)

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

correct

Option 2)

$y=x^{2}+constant$

incorrect

Option 3)

$y^{2}=x+constant$

incorrect

Option 4)

$xy=constant$

incorrect

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