Question

1. Focus is at (0,7) and the Directrix is y = 1.
a) Sketch the parabola.
b) What are the coordinates of the vertex?

Answer 1

Answer:

a) see the attached file

b) Coordinates of vertex = (0, 4)

Step-by-step explanation:

Given:

Focus of parabola is $(0,7)$ and the directrix is $y=1$

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

Here, $(h,k)$ denotes the vertex of parabola, focus is $(h,k+p)$ and the directrix is $y=k-p$

According to question,

$(h,k+p)=(0,7)$

$k-p=1$

So, $h=0$

$k+p=7\,,\,k-p=1$

On adding these equations,

$k+p+k-p=7+1\\2k=8\\k=4$

Put $k=4$ in $k+p=7$

$4+p=7\\p=7-4\\p=3$

So, equation of parabola is $(x-0)^2=4(3)(y-4)\Rightarrow x^2=12(y-4)$

For part a), see the attached file.

b)

Coordinates of vertex = $(h,k)=(0,4)$