In Exercises 1–18, graph each ellipse and locate the foci. (x^2)/(25)+(y^2)/(64)=1

Name: Problem 5, Chapter 9: Precalculus
Uploaded: 2020-06-11T13:58:38-08:00
Duration: 5 min 34 s
Channel: Carrie Hicks
Description: In Exercises 1–18, graph each ellipse and locate the foci. (x^2)/(25)+(y^2)/(64)=1

step 1 We're going to graph the ellipse x squared over 25 plus y squared over 64 equals 1. We know that

In Exercises 1–18, graph each ellipse and locate the foci....

Key Concepts

Foci of an Ellipse

The foci are two fixed points located along the major axis of the ellipse. Their positions are derived using the relationship c = ?(a² - b²) (when a > b), where c is the distance from the center to each focus. The concept of foci is fundamental as it is directly related to the definition and properties of the ellipse.

Graphing Techniques for Ellipses

Graphing an ellipse involves identifying the center, plotting the vertices along the major axis, and the co-vertices along the minor axis. These elements are then used to sketch the elliptical curve accurately, followed by marking the foci. This systematic approach is crucial for a clear visualization of the ellipse’s geometry.

Ellipse

An ellipse is a set of points in a plane where the sum of the distances from each point to two fixed points (called foci) is constant. This geometric shape is characterized by its symmetry, a defined center, and distinct major and minor axes.

Standard Equation of an Ellipse

The standard form of an ellipse’s equation is expressed either as (x - h)²/a² + (y - k)²/b² = 1 for a horizontal ellipse or (x - h)²/b² + (y - k)²/a² = 1 for a vertical ellipse, where (h, k) represents the center, a is the length of the semi-major axis, and b is the length of the semi-minor axis. This form is essential for identifying the key features of the ellipse.

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