Question

A parabola can be drawn given a focus of (-7,3) and a directrix of x = 9. What canbe said about the parabola?

Answer 1

The focus of a parabola is given by:

`F(h,k+p)`

and the directrix is given by:

`y=k-p`

since the directrix is x = 9, we can conclude it is a horizontal parabola, so:

`\begin{gathered} x=9=k-p \\ so\colon \\ k=9+p \end{gathered}`
`\begin{gathered} F(-7,3)=(h,k+p) \\ h=-7 \\ k+p=3 \\ 9+p+p=3 \\ 9+2p=3 \end{gathered}`

solve for p:

`\begin{gathered} 2p=3-9 \\ 2p=-6 \\ p=-(6)/(2) \\ p=-3 \end{gathered}`
`\begin{gathered} k=3-p \\ k=3-(-3) \\ k=6 \end{gathered}`

We can write the parabola in its vertex form:

`\begin{gathered} x=(1)/(4p)(y-k)^2+h \\ so\colon \\ x=-(1)/(12)(y-6)^2-7 \end{gathered}`

It is a horizontal parabola that opens to the left, and has vertex located at (-7,6)