Two short electric dipoles A and B having dipole moment p_1 and p_2...

Let the distance of point $x$ from the origin $O$ be $r$.

Point $x$ lies on the axial line of dipole A. The electric field at $x$ due to dipole A is directed along the horizontal axis and its magnitude is given by:
$E_A = \dfrac{1}{4\pi\epsilon_0} \dfrac{2p_1}{r^3}$

Point $x$ lies on the equatorial line of dipole B. The electric field at $x$ due to dipole B is directed perpendicular to the horizontal axis and its magnitude is given by:
$E_B = \dfrac{1}{4\pi\epsilon_0} \dfrac{p_2}{r^3}$

The resultant electric field makes an angle of $60^{\circ}$ with the line $Ox$ (the horizontal axis). Therefore, the tangent of this angle is the ratio of the perpendicular component of the electric field to the parallel component:
$\tan 60^{\circ} = \dfrac{E_B}{E_A}$

Substituting the expressions for $E_A$ and $E_B$:
$\sqrt{3} = \dfrac{\dfrac{1}{4\pi\epsilon_0} \dfrac{p_2}{r^3}}{\dfrac{1}{4\pi\epsilon_0} \dfrac{2p_1}{r^3}}$

$\sqrt{3} = \dfrac{p_2}{2p_1}$

$\dfrac{p_2}{p_1} = 2\sqrt{3}$

Answer: $2\sqrt{3}$