A short electric dipole is oriented along $+vex$ - direction at origin. At which of following point the electric field have no $x$ component ?

The correct option is C $(1,\sqrt{2},0)$
Consider a point at a distance $r$ from the origin.

Electric field due to dipole $P$ along the direction of position vector $\hat{r}$,
$E_r=\frac{2kP\cos \theta }{r^3}$
Electric field due to dipole $P$ perpendicular to the direction of position vector $\hat{r}$ i.e. $\hat{\theta }$,
$E_{\theta }=\frac{kP\sin \theta }{r^3}$
Horizontal component of net electric field,
$E_x=E_r\cos \theta -E_{\theta }\sin \theta =\frac{kP}{r^3}[2{\cos }^2\theta -{\sin }^2\theta ]$
$E_x=\frac{kP}{r^3}[2-3{\sin }^2\theta ]$
For the point $=(1,\sqrt{2},0);$ $|\hat{r}|=\sqrt{1^2+(\sqrt{2}{)}^2}=\sqrt{3}$
$\sin \theta =\frac{\sqrt{2}}{\sqrt{3}}$
Hence $E_x=0$