A charged particle goes undeflected in a region containing electric and magnetic fields. It is possible that
i) $\overline{\mathrm{E}}\parallel \overline{\mathrm{B}};\overline{\mathrm{V}}\parallel \overline{\mathrm{E}}$
ii) $\overline{\mathrm{E}}$ is perpendicular to $\overline{\mathrm{B}}$ and both are perpendicular to V
iii) $\overline{\mathrm{v}}$ || $\overline{\mathrm{B}}$ but $\overline{\mathrm{E}}$ is not parallel to $\overline{\mathrm{B}}$
iv) $\overline{\mathrm{E}}$ || $\overline{\mathrm{B}}$ but V is not parallel to $\overline{\mathrm{B}}$

Net force on the particle is zero.

So, velocity of the particle may be in the direction of magnetic and electric fields. That means, both fields are parallel, and velocity is also parallel to the direction of the fields.

Magnetic force acts in perpendicular direction of the velocity. So, if magnetic field, electric field and velocity are mutually perpendicular, net force on the particle may be zero. 

i.e, $\overrightarrow{\mathrm{E}}=-\left(\overrightarrow{\mathrm{v}}\times \overrightarrow{\mathrm{B}}\right)$

In other cases given, there will be net force on the body. So, the path will be deflected from straight line.