Answer:
[tex]x=\frac{1}{12}(y-4)^{2}+2[/tex]
Step-by-step explanation:
In this problem we have that
The vertex and the focus has the same y-coordinate, therefore, is a horizontal parabola
The equation of a horizontal parabola is of the form
[tex]x=a(y-k)^{2}+h[/tex]
where
(h,k) is the vertex
The coordinates of the focus are equal to [tex](h+\frac{1}{4a} ,k)[/tex]
we have in this problem
[tex]focus (5,4)[/tex]
[tex]vertex (2,4)[/tex]
substitute
[tex](h,k)=(2,4)[/tex]
[tex](5,4)=(2+\frac{1}{4a} ,4)[/tex]
so
Find the value of a
[tex]5=2+\frac{1}{4a}\\ \\\frac{1}{4a}=3\\ \\a=\frac{1}{12}[/tex]
The equation of the horizontal parabola is equal to
[tex]x=\frac{1}{12}(y-4)^{2}+2[/tex] ------> open to the right
