Concept:
The valency and oxidation state of an element can often be determined from the formulas of its compounds.
The oxidation state of oxygen is generally
\[
\boxed{-2.}
\]
Similarly, halogens such as fluorine or chlorine generally have an oxidation state of
\[
\boxed{-1.}
\]
The compounds formed by an element help identify its group in the periodic table.
Step 1: Determine the oxidation state of \(E\) in \(EO_{2}\).
Let the oxidation state of \(E\) be \(x\).
Since oxygen has oxidation state \(-2\),
\[
x+2(-2)=0.
\]
Therefore,
\[
x-4=0,
\]
\[
x=+4.
\]
Thus,
\[
\boxed{E\text{ exhibits oxidation state }+4.}
\]
Step 2: Determine the oxidation state of \(E\) in \(EX_{4}\).
Assuming \(X\) is a halogen,
\[
X=-1.
\]
Hence,
\[
x+4(-1)=0.
\]
Therefore,
\[
x=+4.
\]
Again,
\[
\boxed{E\text{ has oxidation state }+4.}
\]
Step 3: Identify the group of the element.
Elements of Group 14 commonly exhibit a valency of four and form compounds such as
\[
CO_2,\;SiO_2,\;CCl_4,\;SiCl_4.
\]
Thus, the unknown element belongs to
\[
\boxed{\text{Group 14}.}
\]
Hence,
\[
\boxed{\textbf{Option (B)}}
\]
is the correct answer.