In Exercises 1–18, graph each ellipse and locate the foci. (x^2)/(49)+(y^2)/(81)=1

Name: Problem 7, Chapter 9: Precalculus
Uploaded: 2020-06-11T14:33:01-08:00
Duration: 5 min 7 s
Channel: Carrie Hicks
Description: In Exercises 1–18, graph each ellipse and locate the foci. (x^2)/(49)+(y^2)/(81)=1

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

Key Concepts

Relationship between a, b, and c

For an ellipse, the distance from the center to each focus, denoted as c, is related to the lengths of the semi-major axis (a) and the semi-minor axis (b) by the equation c² = a² - b². This relationship is crucial for determining the positions of the foci relative to the center.

Major and Minor Axes

The major axis of an ellipse is its longest diameter, while the minor axis is its shortest diameter. These axes determine the orientation and the overall shape of the ellipse and are essential for understanding its geometric properties.

Foci

The foci are two key points located along the major axis of an ellipse. They are defined by a specific distance from the center, and every point on the ellipse maintains a constant sum of distances to these two points. This property is fundamental to the definition of an ellipse.

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane for which the sum of the distances to two fixed points (the foci) is constant. This definition leads to a smooth, closed, and symmetric curve.

Standard Equation of an Ellipse

The standard form of an ellipse centered at the origin is expressed as (x²/a²) + (y²/b²) = 1. This equation highlights the geometric features of the ellipse by representing the lengths of its semi-major (a) and semi-minor (b) axes.

Transcript

00:01 We want to graph the ellipse represented by x squared over 49 plus y squared over 81 equals 1.

00:24 Now i know that the center is at 0 ,0.

00:29 And the reason i know that is because there's not a number added to the x and there's not a number added or subtracted from the y.

00:35 So like if we had x minus 3 here, then we would want to go right three units to get to our center.

00:42 And if we had y plus 4, we would want to go down 4.

00:46 But there's nothing added or subtracted to the x or the y, so i know my center's at 0 .0.

00:52 Now the number under the x squared indicates to us how many units to count horizontally to get to a point on the ellipse.

01:02 And we're going to call that b squared.

01:03 So b squared is 49 tells us that b is plus or minus 7.

01:09 So that means we're going to count seven units to the right and seven units to the left.

01:14 Then the number under the y indicates to us how many units we should count vertically from the center.

01:20 We're going to call that a squared.

01:23 And so a squared equals 81, which tells us that the a is plus or minus 9.

01:32 So from my center, let's count horizontally.

01:37 1, 2, 3, 4, 5, 6, 7.

01:45 And we'll count horizontally the other way.

01:49 1, 2, 3, 4, 5, 6, 7.

01:56 Now let's count vertically 9 units.

01:58 1, 2, 3, 4, 5, 6, 7.

02:03 8, 9...

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