Question

Write the equation of the parabola in standard form that satisfies the given conditions. Show all work.Focus: (7, -2)Vertex:(5, -2)

Answer 1

Notice that the focus and the vertex of the parabola are on the line:

`y=-2.`

Therefore the parabola is a horizontal parabola.

Then, the directrix of the parabola is:

`x=5-(7-5)=5-2=3.`

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:

`x-3=\sqrt[]{(x-7)^2+(y-(-2))^2}.`

Then:

`\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}`

Answer:

`(y+2)^2=8(x-5)\text{.}`