Question

Given the directrix x=6 and the focus (3,-5) what is the vertex form of the equation of the parabola?

Answer 1

`x=-(1)/(6)(y+5)^(2)+(9)/(2)`

Step-by-step explanation:

Given a parabola such that:

• The directrix, x=6

,

• Focus = (3, -5)

The standard form of a left-right parabola is given as:

`4p(x-h)=(y-k)^2`

The focus is obtained using the formula:

`\begin{gathered} Focus=(h+p,k) \\ \implies(h+p,k)=(3,-5) \\ \implies h+p=3\cdots(1) \end{gathered}`

The directrix is obtained using the formula:

`\begin{gathered} x=h-p \\ x=6 \\ \implies6=h-p\cdots(2) \end{gathered}`

Solve equations 1 and 2 simultaneously:

Substitute into the standard form:

`\begin{gathered} 4p(x-h)=(y-k)^2 \\ 4(-1.5)(x-4.5)=(y-(-5))^2 \\ -6(x-4.5)=(y+5)^2 \\ x-4.5=-(1)/(6)(y+5)^2 \\ x=-(1)/(6)(y+5)^2+(9)/(2) \end{gathered}`

The vertex form of the parabola is:

`x=-(1)/(6)(y+5)^(2)+(9)/(2)`