find the standard form of the equation of a parabola with the vertex (0,0) and the directrix y=6
find the standard form of the equation of the ellipse with center (0,0) and foci at (+- 4square root 3, 0) and verticies (+-8,0)
To find the standard form of the equation of a parabola and the ellipse, we need to use the formulas and principles related to each shape.
1. Parabola with vertex (0,0) and directrix y=6:
In general, the standard form of a parabola is given by the equation: y^2 = 4px, where (x, y) represents a point on the parabola and p is the distance between the vertex and the focus.
Given that the vertex is (0,0) and the directrix is y=6, we can determine that the focus is located at (0, -p). Since the directrix is y=6, the distance between the vertex and the focus (p) is equal to the distance between the vertex and the directrix.
Using the distance formula, we can calculate p as follows:
distance = sqrt((y2 - y1)^2 + (x2 - x1)^2)
distance = sqrt((6 - 0)^2 + (0 - 0)^2)
distance = sqrt(6^2)
distance = 6
So, p = 6.
Now, substituting the value of p in the standard form equation, we get:
y^2 = 4 * 6 * x
y^2 = 24x
Therefore, the standard form of the equation of the parabola is y^2 = 24x.
2. Ellipse with center (0,0), foci at (+- 4√3, 0), and vertices at (+-8,0):
The standard form of the equation for an ellipse is: (x - h)^2 / a^2 + (y - k)^2 / b^2 = 1, where (h, k) represents the center of the ellipse, and a and b are values related to the size and shape of the ellipse.
In this case, since the center is (0,0), we have h = 0 and k = 0. Also, we are given the foci at (+- 4√3, 0), so the distance from the center to each focus is equal to c = 4√3. Additionally, the vertices are at (+-8,0), so the distance from the center to each vertex is equal to a = 8.
Using the formula, we can calculate b using the relationship between a, b, and c in an ellipse:
b^2 = a^2 - c^2
b^2 = 8^2 - (4√3)^2
b^2 = 64 - 48
b^2 = 16
Therefore, the equation of the ellipse in standard form is:
x^2 / 64 + y^2 / 16 = 1.
To find the standard form of the equation of a parabola with the vertex (0,0) and the directrix y=6, you can use the formula:
(x - h)^2 = 4p(y - k),
where (h, k) represents the vertex and p represents the distance between the vertex and focus/directrix.
In this case, the vertex is (0,0), which means h = 0 and k = 0.
The directrix is y = 6, which means the distance between the vertex and directrix is p = -6.
Substituting these values into the formula, we have:
(x - 0)^2 = 4(-6)(y - 0),
x^2 = -24y.
Therefore, the standard form of the equation of the parabola is x^2 = -24y.
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To find the standard form of the equation of the ellipse with center (0,0), foci at (+- 4√3, 0), and vertices (+-8,0), you can use the formula:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1,
where (h, k) represents the center, a represents the distance from the center to the vertices, and b represents the distance from the center to the foci.
In this case, the center is (0,0), which means h = 0 and k = 0.
The foci are (+- 4√3, 0), which means the distance from the center to the foci is b = 4√3.
The vertices are (+-8, 0), which means the distance from the center to the vertices is a = 8.
Substituting these values into the formula, we have:
(x - 0)^2 / 8^2 + (y - 0)^2 / (4√3)^2 = 1,
x^2 / 64 + y^2 / 48 = 1.
Therefore, the standard form of the equation of the ellipse is x^2 / 64 + y^2 / 48 = 1.