We realize that the vertex is \[\left( 0,\text{ }0 \right)\]and the hub of the parabola is the \[y-axis\]

The condition of the parabola is both of the from \[{{x}^{2~}}=\text{ }4ay\text{ }or\text{ }{{x}^{2}}~=\text{ }-4ay\].

Considering that the parabola goes through point \[\left( 5,\text{ }2 \right)\]which lies in the principal quadrant.

Along these lines, the condition of the parabola is of the structure\[{{x}^{2}}~=\text{ }4ay\], while point \[\left( 5,\text{ }2 \right)\]should fulfill the condition \[~{{x}^{2}}~=\text{ }4ay.\]

Then, at that point,

\[\begin{array}{*{35}{l}}

{{5}^{2}}~=\text{ }4a\left( 2 \right)  \\

25\text{ }=\text{ }8a  \\

a\text{ }=~25/8  \\

\end{array}\]

Consequently, the condition of the parabola is

\[\begin{array}{*{35}{l}}

{{x}^{2}}~=\text{ }4\text{ }\left( 25/8 \right)y  \\

{{x}^{2}}~=\text{ }25y/2  \\

2{{x}^{2}}~=\text{ }25y  \\

\end{array}\]

∴ The condition of the parabola is \[2{{x}^{2}}~=\text{ }25y\]