Answer:
[tex]x=-\frac{1}{6}(y+5)^{2}+\frac{9}{2}[/tex]
Explanation:
Given a parabola such that:
• The directrix, x=6
,
• Focus = (3, -5)
The standard form of a left-right parabola is given as:
[tex]4p(x-h)=(y-k)^2[/tex]
The focus is obtained using the formula:
[tex]\begin{gathered} Focus=(h+p,k) \\ \implies(h+p,k)=(3,-5) \\ \implies h+p=3\cdots(1) \end{gathered}[/tex]
The directrix is obtained using the formula:
[tex]\begin{gathered} x=h-p \\ x=6 \\ \implies6=h-p\cdots(2) \end{gathered}[/tex]
Solve equations 1 and 2 simultaneously:
[tex]\begin{gathered} \begin{equation*} h+p=3\cdots(1) \end{equation*} \\ h-p=6\operatorname{\cdots}(2) \\ Add \\ 2h=9 \\ h=4.5 \\ \text{ Solve for p} \\ 4.5+p=3 \\ p=3-4.5 \\ p=-1.5 \end{gathered}[/tex]
Substitute into the standard form:
[tex]\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=-\frac{1}{6}(y+5)^2 \\ x=-\frac{1}{6}(y+5)^2+\frac{9}{2} \end{gathered}[/tex]
The vertex form of the parabola is:
[tex]x=-\frac{1}{6}(y+5)^{2}+\frac{9}{2}[/tex]
