A short electric dipole is kept at $(0,b).$  Dipole moment of the dipole is  $P\widehat{i}.$ A point charge  $Q$ lies at the origin $\left(O\right).$  Electric field at a point  $A(a,a+b)$ is zero. Find the ratio  $\frac{b}{a}.$

The field at A due to the dipole and point charge Q must be equal and opposite. Field at A due to the dipole makes an angle  $\alpha$ with the line CA, where $\tan \alpha =\frac{1}{2}\tan \theta$
 
 
 
The field due to the dipole must be along OA. A negative charge (Q) at O can create an equal field along AO at point A.
 $\beta =\theta +\alpha$
 $\Rightarrow \tan \beta =\tan (45°+\alpha )$
 $\Rightarrow \frac{a+b}{a}=\frac{1+\frac{1}{2}}{1-1\times \frac{1}{2}}$
 $\Rightarrow 2a=b$