Write the equation of the parabola in standard and general form whose vertex is at the origin and directrix of equation y = 3. Find the coordinate of the focus, length of the latus rectum (focal width), the x- and y-intercepts.
recall that the parabola
x^2 = 4py
has
vertex at (0,0)
directrix is the line y = -p
so that means our parabola has p = -3, giving us
x^2 = -12y
focus is (0,p) = (0,-3)
latus rectum is 4|p| = 12
To find the equation of the parabola, let's first determine the focal length and the equation of the directrix.
1. Focal Length:
The focal length is the distance from the vertex to the focus. Since the vertex is at the origin, the focal length is the perpendicular distance from the origin to the directrix line.
The equation of the directrix is given as y = 3, which is a horizontal line parallel to the x-axis.
The perpendicular distance from the origin (0, 0) to the directrix y = 3 is the absolute value of the y-coordinate, which is 3 units.
Therefore, the focal length (f) is 3 units.
2. Equation of the Directrix:
For a parabola with a horizontal directrix, the equation of the directrix is y = ± f.
In this case, the directrix equation is y = -3.
3. Standard Form of the Parabola:
The standard form of the equation for a parabola with a horizontal directrix is:
(x - h)^2 = 4f(y - k)
The vertex of this parabola is at the origin (0, 0), so h = k = 0.
Using the focal length, we can substitute f = 3 into the standard form equation:
x^2 = 4(3y)
x^2 = 12y
Thus, the equation of the parabola in standard form is x^2 = 12y.
4. General Form of the Parabola:
To convert the equation to general form, we expand the equation:
x^2 = 12y
x^2 - 12y = 0
So, the equation of the parabola in general form is x^2 - 12y = 0.
5. Focus of the Parabola:
The focus of a parabola is located at (h, k + f) in standard form. In this case, since the vertex is at the origin, the focus is at (0, 0 + 3) = (0, 3).
Hence, the coordinate of the focus is (0, 3).
6. Length of the Latus Rectum (Focal Width):
The length of the latus rectum, or the focal width, is 4f units. In this case, 4f = 4 * 3 = 12 units.
So, the length of the latus rectum is 12 units.
7. X- and Y-Intercepts:
To find the x-intercept, we set y = 0 in the equation of the parabola:
x^2 = 12 * 0
x^2 = 0
x = 0
Hence, the x-intercept is (0, 0).
To find the y-intercept, we set x = 0 in the equation of the parabola:
(0)^2 = 12y
0 = 12y
y = 0
Therefore, the y-intercept is (0, 0).
In summary:
- Equation of the Parabola in Standard Form: x^2 = 12y
- Equation of the Parabola in General Form: x^2 - 12y = 0
- Coordinate of the Focus: (0, 3)
- Length of the Latus Rectum (Focal Width): 12 units
- X-Intercept: (0, 0)
- Y-Intercept: (0, 0)
To find the equation of the parabola with a vertex at the origin and a directrix of y = 3, we can use the standard form of the equation of a parabola with its vertex at the origin:
Standard Form: (x - h)^2 = 4p(y - k)
Where the vertex is (h, k) and p is the distance between the vertex and the focus.
In this case, the vertex is at the origin (0, 0), so h = 0 and k = 0. The directrix is given as y = 3, which means the distance from the vertex to the directrix is 3. Since the parabola opens upward, p will be a negative value.
Using the formula, we have:
(x - 0)^2 = 4p(y - 0)
x^2 = 4py
Now we need to find the value of p, which is the distance between the vertex and the focus. The distance between the vertex and the directrix is equal to the distance between the vertex and the focus. So, the distance is 3 units.
p = -3
Substituting the value of p in the equation, we get:
x^2 = -12y
This is the equation of the parabola in standard form.
Now, let's find the coordinates of the focus. The focus lies on the axis of symmetry, which is the x-axis in this case since the parabola opens upward. The y-coordinate of the focus is p units above the vertex.
The focus has coordinates (0, p), so the coordinates of the focus are (0, -3).
Next, let's find the length of the latus rectum (focal width). The latus rectum is the chord passing through the focus and perpendicular to the axis of symmetry. Its length is equal to the absolute value of 4p.
Length of latus rectum = |4p| = |4(-3)| = 12 units
Now, to find the x-intercepts, we set y = 0 in the equation of the parabola and solve for x:
x^2 = -12(0)
x^2 = 0
x = 0
So, the x-intercept is (0, 0).
Finally, to find the y-intercept, we set x = 0 in the equation of the parabola and solve for y:
(0)^2 = -12y
0 = -12y
y = 0
So, the y-intercept is (0, 0).
In summary:
- The equation of the parabola in standard form is x^2 = -12y.
- The coordinates of the focus are (0, -3).
- The length of the latus rectum is 12 units.
- The x-intercept is (0, 0).
- The y-intercept is (0, 0).