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Answer :
Final answer:
The equation of the parabola with focus at (8,16) and directrix at y= -8 is (x - 8)^2 = 48(y - 4), given that the vertex is at (8, 4) and the focal length is 12.
Explanation:
To find the equation of the parabola with the focus at (8,16) and the directrix at y= -8, we can start by noting that the vertex of the parabola will be midway between the focus and directrix. Given that the directrix is 8 feet below the starting point and the focus is 16 feet above, the vertex is at (8, 4). Since this parabola opens upward and the distance between the focus and directrix is 24 feet, the focal length (distance from the vertex to the focus or to the directrix) is 12 feet.
The standard form of a vertical parabola with vertex at (h, k) is given by the equation (x-h)^2 = 4p(y-k), where p is the focal length and (h, k) is the vertex. Substituting the vertex (8, 4) and focal length 12 into this formula, we get:
(x - 8)^2 = 4 * 12 * (y - 4)
Simplifying this, we get:
(x - 8)^2 = 48(y - 4)
Therefore, the equation of the parabola is:
(x - 8)^2 = 48(y - 4).
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Final answer:
To determine the equation of the parabola, use the properties of a parabola and its equation. The equation of the parabola is x^2 - 16x - 16y + 256 = 0.
Explanation:
To determine the equation of the parabola, we need to understand the properties of a parabola and its equation. A parabola is a U-shaped curve that can be formed by a quadratic equation of the form y = ax^2 + bx + c. The focus of the parabola is a point on the axis of symmetry, and the directrix is a line perpendicular to the axis of symmetry.
In this case, the focus is at (8,16) and the directrix is the line y = -8. The distance from the focus to the directrix is equal to the distance from the focus to any point on the parabola. Using this information, we can find the equation of the parabola.
First, we need to find the distance from the focus to the directrix. The distance is 16 units (from the focus to the ceiling) + 8 units (from the floor to the starting point) + 8 units (from the starting point to the directrix) = 32 units.
Since the parabola is equidistant from the floor and the pivot point, the distance from the focus to any point on the parabola is half of the distance from the focus to the directrix. Therefore, the distance from the focus to any point on the parabola is 32/2 = 16 units.
Next, we can use the distance formula to find the equation of the parabola. Let (x, y) be any point on the parabola. The distance from (x, y) to the focus (8, 16) is equal to the distance from (x, y) to the directrix y = -8.
The distance formula is given by:
sqrt((x - 8)^2 + (y - 16)^2) = |y - (-8)|
Squaring both sides of the equation, we get:
(x - 8)^2 + (y - 16)^2 = (y + 8)^2
Expanding and simplifying the equation, we get:
x^2 - 16x + 64 + y^2 - 32y + 256 = y^2 + 16y + 64
Combine like terms:
x^2 - 16x + 256 - 16y = 0
The equation of the parabola is x^2 - 16x - 16y + 256 = 0.
