Answer:
[tex](x+7)^2=9(y-2)[/tex]
Step-by-step explanation:
As the directrix is horizontal (y = -1/4), the parabola is vertical.
The standard formula of a vertical parabola is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Standard form of a vertical parabola}}\\\\(x-h)^2=4p(y-k)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{Vertex:\;$(h,k)$}\\\phantom{ww}\bullet\;\textsf{Focus:\;$(h,k + p)$}\\\phantom{ww}\bullet\;\textsf{Directrix:\;$y=(k-p)$}\\\phantom{ww}\bullet\;\textsf{Axis of symmetry:\;$x=h$}\\\phantom{ww}\bullet\;\textsf{$p =$ distance between vertex and focus.}\end{array}}[/tex]
The vertex of a vertical parabola shares the same x-coordinate as the focus and lies equidistant between the focus and the directrix.
Given that the focus is at (-7, 17/4) and the directrix is y = -1/4, the coordinates of the vertex are:
[tex](h,k)=\left(-7, \dfrac{\frac{17}{4}-\frac{1}{4}}{2}\right)\\\\\\(h,k)=\left(-7, \dfrac{4}{2}\right)\\\\\\(h,k)=(-7,2)[/tex]
The value of p is the distance between the vertex and the focus.
Given that the focus is (-7, 17/4) and the vertex is (-7, 2), then p is the difference between the y-coordinates:
[tex]p = \dfrac{17}{4}-2=\dfrac{9}{4}[/tex]
Now, substitute h = -7, k = 2 and p = 9/4 into the formula and simplify:
[tex](x - (-7))^2=4\cdot \dfrac{9}{4}(y-2)\\\\\\(x+7)^2=9(y-2)[/tex]
Therefore, the equation of the parabola is:
[tex]\Large\boxed{\boxed{(x+7)^2=9(y-2)}}[/tex]
