1. Electric field of a dipole on axial line

For an electric dipole of moment $p$, at a point on the axial line at distance $r$ from the center, the electric field is

$E_{\text{axial}}=\frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3}$

Given that the electric field at point $P$ is $E$, we take

$E=\frac{1}{4\pi\varepsilon_0}\frac{2p}{r^3}$

2. Electric field of a dipole on equatorial line

At a point on the equatorial line at the same distance $r$ from the center, the magnitude of electric field is

$E_{\text{equatorial}}=\frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}$

3. Compare the two fields

So,

$\frac{E_{\text{equatorial}}}{E}=\frac{\dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3}}{\dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}}=\frac{1}{2}$

Hence,

$E_{\text{equatorial}}=\frac{E}{2}$

Therefore,

$\frac{E}{x}=\frac{E}{2} \implies x=2$

4. Comparison with stored answer

The derived answer is $2$, not $16$.

So the stored correct answer appears to be incorrect unless some figure or extra distance relation is missing from the question.