Answer:
y = - x²/12
Step-by-step explanation:
The axis of symmetry passes through both the focus and vertex, and both of those are on the y-axis. So, the y- axis *is* the axis of symmetry. The equation has the form y = ax^2 and since the focus is *below* the vertex, the parabola opens downward and the coefficient a is negative.
The vertex is halfway between the focus and the nearest point on the directrix, so the focal length (distance from vertex to focus is) 6/2 = 3. The signed focal distance is f = -3, since it's below the vertex.
A point (x,y) is on the parabola if it's equidistant from the line y=3 and the focus F=(0,-3).
|y - 3| = √[(x - 0)² + (y + 3)²] ..... distance to directrix = distance to F
(y - 3)² = x² + (y + 3)²
y² - 6x + 9 = x² + y² + 6y + 9
-12y = x²
y = - x²/12
The equation of the parabola that has its vertex at the origin, focus on the negative y-axis, 6 units away from the directrix is [tex]\mathbf{x^2 = -12y}[/tex]
The equation of a parabola is represented as:
[tex]\mathbf{x^2 = 4py}[/tex]
Where:
[tex]\mathbf{F = (0,p)}[/tex] -- focus
[tex]\mathbf{y = -p}[/tex] --- the directrix
From the question, the focus is on the negative y-axis,
So, we rewrite the focus as:
[tex]\mathbf{F = (0,p)}[/tex]
Where:
[tex]\mathbf{p < 0}[/tex]
The distance between the focus and the directrix is represented as:
[tex]\mathbf{|p - (-p)| = 6}[/tex]
Remove bracket
[tex]\mathbf{|p +p| = 6}[/tex]
Add
[tex]\mathbf{|2p| = 6}[/tex]
Rewrite as:
[tex]\mathbf{2|p| = 6}[/tex]
Divide both sides by 2
[tex]\mathbf{|p| = 3}[/tex]
Remove absolute bracket
[tex]\mathbf{p = 3\ or\ p = -3}[/tex]
Recall that: [tex]\mathbf{p < 0}[/tex]
So, we have:
[tex]\mathbf{p = -3}[/tex]
Substitute -3 for p in [tex]\mathbf{x^2 = 4py}[/tex]
[tex]\mathbf{x^2 = 4(-3)y}[/tex]
[tex]\mathbf{x^2 = -12y}[/tex]
Hence, the equation of the parabola is [tex]\mathbf{x^2 = -12y}[/tex]
Read more about parabolas at:
https://brainly.com/question/4443998
