Answer:
[tex]x^2 = 2y[/tex] --- equation
[tex](x,y) = (0,\frac{1}{2})[/tex] --- focus
[tex]y = -\frac{1}{2}[/tex] --- directrix
[tex]Width = 2[/tex] ---- focal width
Step-by-step explanation:
Given
[tex]depth = 2[/tex]
[tex]diameter = 4[/tex]
Required
The equation of parabola
The depth represents the y-axis. So:
[tex]y = 2[/tex]
The diameter represents how the parabola is evenly distributed across the x-axis.
We have:
[tex]diameter = 4[/tex]
-2 to 2 is 4 units.
So:
[tex]x = [-2,2][/tex]
So, the coordinates of the parabola is:
[tex](-2,2)\ and\ (2,2)[/tex]
The equation of the parabola is calculated using:
[tex]x^2 = 4py[/tex]
Substitute (-2,2) for (x,y)
[tex](-2)^2 = 4p*2[/tex]
[tex]4 = 8p[/tex]
Divide by 8
[tex]p = \frac{4}{8}[/tex]
[tex]p = \frac{1}{2}[/tex]
So, the equation is:
[tex]x^2 = 4py[/tex]
[tex]x^2 = 4 * \frac{1}{2} * y[/tex]
[tex]x^2 = 2y[/tex]
The defining features
(a) Focus
The focus is located at:
[tex](x,y) = (0,p)[/tex]
[tex](x,y) = (0,\frac{1}{2})[/tex]
(b) Directrix (y)
[tex]y = -p[/tex]
[tex]y = -\frac{1}{2}[/tex]
(c) Focal width
[tex]Width = 4p[/tex]
[tex]Width = 4*\frac{1}{2}[/tex]
[tex]Width = 2[/tex]
