Notice that the focus and the vertex of the parabola are on the line:
[tex]y=-2.[/tex]
Therefore the parabola is a horizontal parabola.
Then, the directrix of the parabola is:
[tex]x=5-(7-5)=5-2=3.[/tex]
Now, recall that a parabola is a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix).
Therefore, (x,y) is on the graph of the parabola if:
[tex]x-3=\sqrt[]{(x-7)^2+(y-(-2))^2}.[/tex]
Then:
[tex]\begin{gathered} (x-3)^2=(x-7)^2+(y+2)^2, \\ (x-3)^2-(x-7)^2=(y+2)^2, \\ (x-3+x-7)(x-3-x+7)=(y+2)^2, \\ (2x-10)(4)=(y+2)^2, \\ 8(x-5)=(y+2)^2\text{.} \end{gathered}[/tex]
Answer:
[tex](y+2)^2=8(x-5)\text{.}[/tex]
