In Exercises 1-18, graph each ellipse and locate the foci. (x^2)/(25)+(y^2)/(16)=1

Name: Problem 2, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-05T13:33:41-08:00
Duration: 1 min 30 s
Channel: Ankit Gupta
Description: In Exercises 1-18, graph each ellipse and locate the foci. (x^2)/(25)+(y^2)/(16)=1

step 1 We have given that x squared divided by 25 plus y squared divided by 16 is equals to 1. We can r

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

Key Concepts

Ellipse

An ellipse is the set of all points in a plane for which the sum of the distances from two fixed points (the foci) is constant. It is a conic section that exhibits symmetry about both its major and minor axes.

Standard Equation of an Ellipse

The standard form of the equation 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-major and semi-minor axes respectively. This form makes it easier to identify the orientation and dimensions of the ellipse.

Semi-Major and Semi-Minor Axes

The semi-major axis is the longest radius of the ellipse, and the semi-minor axis is the shortest. Their values determine the shape and size of the ellipse. By comparing the denominators in the standard equation, one can determine which axis is longer and, consequently, the orientation of the ellipse (horizontal or vertical).

Foci Calculation

For an ellipse, the distance from the center to each focus is given by c, where c² = a² - b² for a horizontal ellipse (or c² = b² - a² for a vertical ellipse). The foci always lie along the major axis and are used to define the locus of points that form the ellipse.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, determining the lengths and directions of the semi-major and semi-minor axes, locating the vertices and co-vertices, and marking the foci. This visualization helps in understanding the geometric properties of the ellipse, such as its symmetry and eccentricity.

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