Answer:
a) see the attached file
b) Coordinates of vertex = (0, 4)
Step-by-step explanation:
Given:
Focus of parabola is [tex](0,7)[/tex] and the directrix is [tex]y=1[/tex]
To sketch: the parabola
To find: coordinates of the vertex
Solution:
As the focus lies on the y-axis and y-coordinate of the focus is positive,
equation of parabola is [tex](x-h)^2=4p(y-k)[/tex]
Here, [tex](h,k)[/tex] denotes the vertex of parabola, focus is [tex](h,k+p)[/tex] and the directrix is [tex]y=k-p[/tex]
According to question,
[tex](h,k+p)=(0,7)[/tex]
[tex]k-p=1[/tex]
So, [tex]h=0[/tex]
[tex]k+p=7\,,\,k-p=1[/tex]
On adding these equations,
[tex]k+p+k-p=7+1\\2k=8\\k=4[/tex]
Put [tex]k=4[/tex] in [tex]k+p=7[/tex]
[tex]4+p=7\\p=7-4\\p=3[/tex]
So, equation of parabola is [tex](x-0)^2=4(3)(y-4)\Rightarrow x^2=12(y-4)[/tex]
For part a), see the attached file.
b)
Coordinates of vertex = [tex](h,k)=(0,4)[/tex]
