Ch. 7 - Conic Sections

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Blitzer 8th Edition

Ch. 7 - Conic Sections

Problema 21

Capitolo 8, Problema 21

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, 15); Directrix: y = - 15

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Recall that the standard form of a parabola with a vertical axis of symmetry 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 (or directrix).

Identify the given focus $ (0, 15) $ and directrix $ y = -15 $. The vertex lies exactly halfway between the focus and the directrix on the vertical line $ x = 0 $.

Calculate the vertex coordinates by finding the midpoint between the focus and directrix: $ k = \frac{15 + (-15)}{2} = 0 $, so the vertex is at $ (0, 0) $.

Determine the value of $ p $, which is the distance from the vertex to the focus (or directrix). Since the focus is at $ y = 15 $ and the vertex at $ y = 0 $, $ p = 15 $.

Write the equation of the parabola using the vertex form: $ (x - 0)^2 = 4 \times 15 \times (y - 0) $, which simplifies to $ x^2 = 60y $. This is the standard form of the parabola.

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Definition of a Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. Understanding this definition helps in deriving the equation of the parabola by using the distance formula between any point on the parabola and the focus and directrix.

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Horizontal Parabolas

Standard Form of a Parabola

The standard form of a parabola's equation depends on its orientation. For a vertical parabola with vertex at (h, k), the form is (x - h)^2 = 4p(y - k), where p is the distance from the vertex to the focus or directrix. This form is essential for writing the equation once the vertex and p are known.

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Parabolas as Conic Sections

Finding the Vertex and Parameter p

The vertex lies midway between the focus and directrix. The parameter p is the distance from the vertex to the focus (positive if the parabola opens upward, negative if downward). Calculating these values allows you to write the parabola's equation in standard form accurately.

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Vertex Form

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