Question

Writing an equation of a parabola given the vertex and the focus

Answer 1

vertex (3, 1)

directrix y = 6

The equation of a parabola is

`y=(1)/(4(f-k))(x-h)^2+k`

where,

(h,k) is the vertex and (h,f) is the focus

Thus,

h = 3

k = 1

The distance from the focus to the vertex is equal to the distance from the vertex to the directrix, then f - k = k - 6

replace k=1 and solve for f,

`\begin{gathered} f-1=1-6 \\ f=-5+1 \\ f=-4 \end{gathered}`

Thus,

h = 3

k = 1

f = -4

therefore, the equation of the parabola is,

`\begin{gathered} y=(1)/(4*(-4-1))(x-3)^2+1 \\ \\ y=-(1)/(20)(x-3)^(2)+1 \end{gathered}`