In Exercises 1-18, graph each ellipse and locate the foci. (x^2)/(64)+(y^2)/(100)=1

Name: Problem 8, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-05T13:37:55-08:00
Duration: 1 min 19 s
Channel: Ankit Gupta
Description: In Exercises 1-18, graph each ellipse and locate the foci. (x^2)/(64)+(y^2)/(100)=1

step 1 I have given an equation of ellipse which is x squared divided by 64 plus y square divided by 10

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

Key Concepts

Foci of an Ellipse

The foci are two fixed points along the major axis of the ellipse and are located at a distance c from the center, where c is calculated using the relationship c² = |a² - b²|. The positions of the foci are essential in defining the ellipse, as they are used in the geometric definition that makes the ellipse unique.

Major and Minor Axes

In the equation of an ellipse, the larger of the two values between a and b corresponds to the semi-major axis, while the smaller value corresponds to the semi-minor axis. The major axis determines the primary direction in which the ellipse is elongated, and this information is crucial for locating key features like the foci.

Ellipse

An ellipse is a set of all points in a plane such that the sum of their distances from two fixed points (the foci) is constant. It represents a conic section that occurs when a plane intersects a cone at a shallow angle, and its shape is characterized by a smooth, closed curve that is longer in one direction than the other.

Standard Equation of an Ellipse

The standard form of an ellipse centered at the origin is given by (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-axes. This form clearly identifies the lengths of the major and minor axes and simplifies the process of graphing the ellipse and calculating other properties such as the foci.

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