Ch. 7 - Conic Sections

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 7 - Conic Sections

Problem 23

Chapter 8, Problem 23

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, - 25); Directrix: y = 25

Verified step by step guidance

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, -25) $ and directrix $ y = 25 $. The vertex lies exactly halfway between the focus and the directrix along the vertical line.

Calculate the vertex $ (h, k) $ by finding the midpoint between the focus and directrix: $ k = \frac{-25 + 25}{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 vertex is at $ y=0 $ and the focus is at $ y=-25 $, $ p = -25 $ (negative because the parabola opens downward).

Substitute $ h=0 $, $ k=0 $, and $ p=-25 $ into the standard form equation to get $ (x - 0)^2 = 4(-25)(y - 0) $, which simplifies to $ x^2 = -100y $. This is the standard form of the parabola.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 based on the given focus and directrix.

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.

Finding the Vertex from Focus and Directrix

The vertex of a parabola lies midway between the focus and the directrix. By calculating the midpoint between the focus (0, -25) and the directrix y = 25, you can find the vertex, which is crucial for expressing the parabola's equation in standard form.

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