If an object’s acceleration is constant in magnitude and direction, must its path be a straight line? If not, give an example.

What the condition “constant acceleration” really tells you

A constant acceleration vector means $\vec a$ has the same magnitude and points in the same direction at all times. That controls how the velocity changes, but it does not force the velocity direction to stay fixed.

When the path is straight

For the position to lie on a line, the velocity direction must not rotate. That happens when $\vec v$ is always parallel to $\vec a$ (or when $\vec a = \vec 0$), so the motion stays along a single line.

How you can still get a curve

Suppose $\vec a$ points straight down, but the object initially has some sideways velocity. The downward acceleration changes the vertical component of velocity, while the horizontal component stays constant. Because the velocity direction changes over time, the trajectory bends.

Concrete example: projectile motion

Take $\vec a = -g\,\hat y$ (constant gravity) and initial velocity $\vec v_0 = v_{0x}\,\hat x + v_{0y}\,\hat y$ with $v_{0x} \neq 0$. The motion is $$x(t) = x_0 + v_{0x} t, \quad y(t) = y_0 + v_{0y} t - \tfrac{1}{2} g t^2,$$ and eliminating $t$ gives a parabola: $$y(x) = y_0 + \frac{v_{0y}}{v_{0x}}(x-x_0) - \frac{g}{2v_{0x}^2}(x-x_0)^2.$$ So the path is curved even though $\vec a$ is constant.