An electron projected with velocity $\overrightarrow{\mathrm{v}}={\mathrm{v}}_0\hat{\mathrm{i}}$ in the electric field $\overrightarrow{\mathrm{E}}={\mathrm{E}}_0\widehat{\mathrm{j}}$. Trace the path followed by the electron E₀.

When we see the condition mentioned in the question it says that a particle is projected in an electric field (${\mathrm{E}}_0$) in the j direction which is along the y-axis in such a way that it is moving with a velocity V₀ in vector  direction which is along  the x axis. 

The acceleration of the particle will depend on the mass of the particle projected on the external electric field and the force produced on the particle due to the externally applied field.
The acceleration of the electron will be $-\frac{\mathrm{eE}}{\mathrm{m}}\widehat{\mathrm{j}}.$

Therefore after time t , x = $V_{0}t and y = -\frac{1}{2 }\left(\frac{eE}{m}\right)t^2.$

Eliminating t , y = - $\frac{eE}{2m{V^2}_0}{x}^2.$ This equation represents a parabola .

The motion of such a particle is analogous to the motion of the particle which is projected  horizontally with a velocity $V_0$ with the gravity acting on the particle which is in the vertical direction.
So, the particle will trace along a parabolic path in the x-y plane.