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A particle is moving with a velocity $v = k\left( {yi + xj} \right)$ where $k$ is a constant. The general equation for the path described by the particle is
a. $y = {x^2} + ct$
b. \[{y^2} = x + c\]
c. \[xy = c\]
d. \[{y^2} = {x^2} + c\]

Answer
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Hint: The velocity and the position can be derived from the newton equation by the method of integration. Velocity is the rate of change of displacement.
First find the velocity along x axis then along the y axis.
Then equate the two equations. Then obtain one of the above equations using the above equations.

Formula used:
$v = \dfrac{{dx}}{{dt}}$
$v$ is the velocity and $t$ is the time.

Complete step by step answer:
To describe the apposition of a body, its velocity or acceleration relative to frame of reference we use the kinematic equation.
Velocity is the rate of change of displacement. From the newton equation, velocity is derived by the method of integration. Integration of velocity results in the acceleration equation.
If the motion starts from rest and the frame of reference should be the same, the initial velocity will be zero. If the motion starts from rest and the frame of reference should be the same.
It is a scalar quantity. The body attains uniform motion along a straight line when that body is moving with uniform velocity.

The velocity Displacement may or may not be equal to the path length travelled of an object. Distance to unit time is called speed.
Equation integration results in the distance equation.
Then the velocity along x axis
$ \Rightarrow \dfrac{{dx}}{{dt}} = ky$
Then the velocity along x axis
$ \Rightarrow \dfrac{{dy}}{{dt}} = kx$
Now let us find out
$ \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{\dfrac{{dy}}{{dt}}}}{{\dfrac{{dx}}{{dt}}}} = \dfrac{x}{y}$
On cross multiply the terms and we get
 $ \Rightarrow ydy = xdx$
Now by integration we get
$ \Rightarrow {y^2} = {x^2} + c$

Hence, the correct answer is option (A).

Note: The motion starts from rest and the frame of reference should be the same. If the initial velocity is zero. The velocity equation integration results in the acceleration equation. Displacement may or may not be equal to the path length travelled of an object. Distance to unit time is called speed.