Answer:
Vertex: (1,-4)
Focus: (1,-1)
Directrix: y = -7
Step-by-step explanation:
We have been given the equation of parabola [tex]12y=(x-1)^2 -48[/tex]
Divide both sides by 12 to get the equation in vertex form
[tex]y=\frac{1}{12}(x-1)^2 -4[/tex]
The vertex form of the parabola is
[tex]y=4p(x-h)^2+k[/tex], where (h.k) is the vertex.
On comparing, we get
Vertex = (h,k) = (1,-4)
[tex]\frac{1}{4p}=\frac{1}{12}\\\\p=3[/tex]
Now, we know that focus always lie inside the parabola and directrix is always outside the parabola.
Also, the vertex is the mid point of focus and the point on directrix. All three lie on the same line called axis of symmetry. The distance between focus and vertex is p
Thus, the focus is at 3 units upward from vertex
Focus is given by
[tex](1,-4+3)\\\\=(1,-1)[/tex]
And the directrix is a parallel line to x axis and at a distance of 3 units down from the vertex.
Hence, directrix is
y = -4-3
y = -7
