The general equation represented as: [tex]\mathbf{x^2 = 2py}[/tex]
- The focus is [tex]\mathbf{F = (0,16)}[/tex]
- The directrix is [tex]\mathbf{y = -16}[/tex]
- The axis of symmetry is [tex]\mathbf{x = 0}[/tex]
The equation is given as:
[tex]\mathbf{x^2 = 64y}[/tex]
Recall that:
[tex]\mathbf{x^2 = 2py}[/tex]
Where:
[tex]\mathbf{F = (0,\frac p2)}[/tex] --- the focus
[tex]\mathbf{y = -\frac p2}[/tex] --- the directrix
Compare [tex]\mathbf{x^2 = 2py}[/tex] and [tex]\mathbf{x^2 = 64y}[/tex], we have:
[tex]\mathbf{2py = 64y}[/tex]
Divide both sides by 2y
[tex]\mathbf{p = 32}[/tex]
Recall that:
[tex]\mathbf{F = (0,\frac p2)}[/tex] and [tex]\mathbf{y = -\frac p2}[/tex]
So, we have:
[tex]\mathbf{F = (0,\frac {32}2)}[/tex]
[tex]\mathbf{F = (0,16)}[/tex]
[tex]\mathbf{y = -\frac p2}[/tex]
[tex]\mathbf{y = -\frac{32}{2}}[/tex]
[tex]\mathbf{y = -16}[/tex]
The axis of symmetry is [tex]\mathbf{x = 0}[/tex]
Read more about equations of parabola at:
https://brainly.com/question/11911877
