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Find the standard form of the equation of each ellipse and give the location of its foci.
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Ch. 7 - Conic Sections
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 8, Problema 23
Find the standard form of the equation of each parabola satisfying the given conditions. Focus: (0, - 25); Directrix: y = 25
Guida verificata passo dopo passo1
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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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.
Video consigliato:
5:28
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.
Video consigliato:
5:33Parabolas as Conic Sections
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.
Video consigliato:
7:18Horizontal Parabolas Example 1
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