A particle is moving along the curve x = at² + bt + c. If ac = b², then particle would be moving with uniform

The correct option is acceleration.

Let us analyze the motion of the particle. The position of the particle as a function of time $t$ along the curve (or path) is given by the relation:
$x=a{t}^2+bt+c$
where $a$, $b$, and $c$ are constants.

To find the velocity of the particle, we differentiate the position function $x$ with respect to time $t$:
$v=\frac{dx}{dt}=\frac{d}{dt}\left(a{t}^2+bt+c\right)$
Using the power rule of differentiation, we get:
$v=2at+b$

Now, to find the acceleration of the particle, we differentiate the velocity function $v$ with respect to time $t$:
$A=\frac{dv}{dt}=\frac{d}{dt}\left(2at+b\right)$
Differentiating this, we obtain:
$A=2a$

Since $a$ is a constant, the acceleration $A=2a$ is also a constant. A constant acceleration means that the particle moves with a uniform (non-changing) acceleration. Note that the condition $ac=b^2$ consists of constants, which does not alter the fact that $a$ is constant and therefore the acceleration remains constant.