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8. Conic Sections
Parabolas
2:52 minutesTextbook Question
Find the standard form of the equation of the parabola satisfying the given conditions. Focus: (0,-11); Directrix: y=11
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Identify the key components of the parabola: The focus is (0, -11), and the directrix is y = 11. Since the focus and directrix are vertically aligned, the parabola opens either upward or downward. The vertex will be the midpoint between the focus and the directrix.
Calculate the vertex: The vertex lies halfway between the focus and the directrix. Use the midpoint formula for the y-coordinates: \( y_{vertex} = \frac{y_{focus} + y_{directrix}}{2} \). Substituting \( y_{focus} = -11 \) and \( y_{directrix} = 11 \), find the y-coordinate of the vertex. The x-coordinate of the vertex is the same as the x-coordinate of the focus, which is 0.
Determine the distance \( p \): The distance \( p \) is the distance from the vertex to the focus (or equivalently, from the vertex to the directrix). Use \( p = |y_{focus} - y_{vertex}| \).
Write the standard form of the parabola: For a parabola that opens vertically, the standard form is \( (x - h)^2 = 4p(y - k) \), where \( (h, k) \) is the vertex and \( p \) is the distance calculated in the previous step. Substitute the values of \( h \), \( k \), and \( p \) into the equation.
Simplify the equation: Expand and simplify the equation if necessary to express it in its final standard form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Parabola
The standard form of a parabola that opens vertically is given by the equation (x - h)² = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus. This form allows for easy identification of the vertex and the direction in which the parabola opens, which is crucial for graphing and understanding its properties.
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Parabolas as Conic Sections
Focus and Directrix
In the context of parabolas, the focus is a fixed point from which distances to points on the parabola are measured, while the directrix is a line that is perpendicular to the axis of symmetry of the parabola. The distance from any point on the parabola to the focus is equal to the distance from that point to the directrix, which is fundamental in deriving the equation of the parabola.
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Vertex of the Parabola
The vertex of a parabola is the point where it changes direction and is located midway between the focus and the directrix. For the given conditions, the vertex can be calculated as the midpoint of the focus and the directrix, which is essential for determining the standard form of the parabola's equation.
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