How do you write the equation of the parabola in vertex form given vertex \[\left( 5,-2 \right)\] and focus \[\left( 5,-4 \right)\].

Hint: In this problem, we have to find the equation of the parabola in vertex form from the given vertex \[\left( 5,-2 \right)\] and the focus \[\left( 5,-4 \right)\]. To find the equation of the parabola in vertex form, we know that the parabola is open downwards as the focus point is below the vertex point. We also know that the length of the vertex and focus is a, substituting the value of a and the vertex in the vertex form equation, we can find the equation of the parabola.

Complete step-by-step solution:
We know that the given vertex point is \[\left( 5,-2 \right)\] and the focus is \[\left( 5,-4 \right)\].
we know that the parabola is open downwards as the focus point is below the vertex point.
We know that the equation of the open downwards parabola in vertex form is of the form.
\[{{\left( x-h \right)}^{2}}=-4a\left( y-k \right)\] …….. (1)
Here, the vertex \[\left( h,k \right)\] is \[\left( 5,-2 \right)\].
Now we have to find the length between focus and the vertex, a.
The length between focus and the vertex, a = \[\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}\] …… (2)
 Here \[\left( {{x}_{1}},{{y}_{1}} \right)\] is the focus point \[\left( 5,-4 \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] is the vertex point \[\left( 5,-2 \right)\].
Now we can substitute the above points in (2), we get
\[\begin{align}
  & \Rightarrow a=\sqrt{{{\left( 5-5 \right)}^{2}}+{{\left( -2+4 \right)}^{2}}} \\
 & \Rightarrow a=\sqrt{4}=2 \\
\end{align}\]
The value of a is 2.
Now we can substitute the vertex point \[\left( 5,-2 \right)\] and the value of a in (1), we get
\[\begin{align}
  & \Rightarrow {{\left( x-5 \right)}^{2}}=-4\left( 2 \right)\left( y-\left( -2 \right) \right) \\
 & \Rightarrow {{\left( x-5 \right)}^{2}}=-8\left( y+2 \right) \\
\end{align}\]
Therefore, the equation of parabola in vertex form is \[{{\left( x-5 \right)}^{2}}=-8\left( y+2 \right)\].


Note: Students make mistakes while finding the distance between focus and the vertex which is the value of a. To solve these types of problems, students should understand the concept, formulas and properties of parabola. The general equation should be written correct for the respective parabolas.