In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given

In Exercises 25–36, find the standard form of the equation...

Key Concepts

Ellipse

An ellipse is a 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. It exhibits symmetry about both its major and minor axes, and its shape is determined by the distance between its foci relative to the lengths of its axes.

Standard Form Equation of an Ellipse

The standard form of the equation for an ellipse is written differently depending on the orientation of its major axis. For an ellipse centered at (h, k), if the major axis is horizontal, the equation is ((x - h)² / a²) + ((y - k)² / b²) = 1; if vertical, it is ((x - h)² / b²) + ((y - k)² / a²) = 1, where 'a' is the distance from the center to a vertex along the major axis and 'b' is the distance from the center to a vertex along the minor axis.

Foci

Foci are two distinct points located along the major axis of an ellipse that are used to define it. The fundamental property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant. The distance from the center to each focus is denoted by 'c', and this measure plays a key role in determining the ellipse's eccentricity.

Vertices

Vertices are the points on an ellipse farthest from the center along the major axis. They provide the maximum spread of the ellipse in the direction of the major axis and are used to define the semi-major axis length, 'a'. In constructing the ellipse's equation, knowing the vertices is crucial for establishing its overall dimensions.

Relationship Among a, b, and c

In any ellipse, the relationship between the semi-major axis (a), the semi-minor axis (b), and the focal distance (c) is given by a² = b² + c². This equation links the geometry of the ellipse by allowing one to determine any one of the parameters if the other two are known, highlighting the interconnected nature of the ellipse's dimensions.

Transcript

00:01 We are writing the equation of an ellipse, and what we know about this ellipse is that the fosi are at 0 -9 -4 and 0 -4.

00:15 So 0 -9 -4 and 0 -4.

00:25 We're also told that the vertices are at 0 -9 -6.

00:38 And 07.

00:39 So 0 negative 7 and 07.

00:54 I'm going to draw a picture to help me think about it.

00:59 The vertices are at 0 negative 7, so 1, 2, 3, 4, 5, 6, 7 here, and 0, 7, 7 here.

01:17 And then my fosia, 0 negative 4 here, 1, 2, 3, 4 here.

01:23 Notice that the center will be halfway between the foci.

01:29 It's also halfway between the vertices, and so that will be at 0 .0.

01:37 Now let's start writing our equation.

01:42 We'll have x minus h squared.

01:44 Well, in this case the h is 0.

01:46 The h is the x.

01:47 Coordinate of the center.

01:49 So x squared over, we've got to figure out what number goes under the x squared, plus now we have y minus zero so we can write that as y squared over.

02:01 Now vertically we counted from the center we counted seven units to get to a point on the ellipse.

02:08 And seven squared is forty nine and we know in the standard equation of an ellipse we'll have a one here.

02:14 Now to figure out what number goes under the x squared we're going to use the formula c squared equals a squared minus b squared...

Please give Ace some feedback