Sketching an Ellipse In Exercises 25-30 , find the center, foci, vertices, and eccentricity of t

Sketching an Ellipse In Exercises 25-30 , find the center,...

Key Concepts

Foci of the Ellipse

The foci of an ellipse are two distinct points located along the major axis which collectively determine the shape of the ellipse. They are found using the relationship c² = a² - b², where c is the distance from the center to each focus. The sum of the distances from any point on the ellipse to the foci is constant and equals 2a, defining one of the ellipse’s key geometric properties.

Vertices of the Ellipse

Vertices are the points where the ellipse intersects its major and minor axes. Along the major axis, the vertices are located a units away from the center, and along the minor axis, they are located b units away. These points represent the maximum and minimum distances from the center along the respective axes, which is crucial for graphing the ellipse accurately.

Eccentricity of the Ellipse

The eccentricity of an ellipse, denoted by e, is a measure of how much the ellipse deviates from being circular, defined as the ratio e = c/a, where c is the distance from the center to a focus and a is the length of the semi-major axis. An ellipse with an eccentricity near 0 is nearly circular, while an eccentricity closer to 1 indicates a more elongated shape.

Ellipse Standard Form

The standard form of the ellipse equation allows for easier identification of its key properties by rewriting the equation in the form (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse. This form helps distinguish the lengths of the semi-major axis (a) and semi-minor axis (b), and determines the orientation of the ellipse along the coordinate axes.

Center of the Ellipse

The center of an ellipse is the point around which the ellipse is symmetrically drawn. In the standard form, the center is given by (h, k). When the equation lacks linear x or y terms, the center is at the origin (0, 0), simplifying the analysis of the ellipse's position and symmetry.

Transcript

00:01 To find all the information of the ellipse that we have to find, let's start by rewriting this equation into standard form.

00:10 So right now we start with 3x squared plus 7 y squared equal 63, in standard form for an ellipse has the equation set equal to 1.

00:20 So if we want to get this equation equal to 1, let's divide everything by 63.

00:25 So for this term, when you divide 3x squared by 63, we're going to get x squared over 21.

00:33 And when we divide 7 y squared to 63, we're going to get y squared over 9.

00:42 Alright, so from this equation, this equation is a standard form.

00:50 Since this is x minus h squared and this is y minus k squared, this means h and k is 0.

00:56 And that means that the center of the ellip is at 0 comma 0 at the origin.

01:04 Next we need to find the two vertices, of the ellipse.

01:08 So remember the vertices are in the same direction as the faux side.

01:14 And they're going to be determined by the larger of the two numbers here.

01:20 So this is our a, since it's bigger, that's our b, since it's smaller.

01:24 With a, it is a square root of 21, since a squared of 21.

01:30 And let's also put down b, b is just 3, it's square root of 9.

01:38 This a here is the distance from the center to the two vertices of the ellipse.

01:44 Since it's 21 under x squared, that means it's in the x direction.

01:48 In other words, left and right.

01:49 So the two vertices are going to be square of 21 comma 0 and negative square of 21 comma 0.

02:00 So then left to the right and the left of the center.

02:05 We have the center.

02:07 We have the vertices.

02:09 We also need to find the both sides.

02:11 To find the faux side, we need to find c.

02:13 C is the distance from the center to the two both sides.

02:18 So to find that, we're going to use this equation, c squared equals a squared minus b squared.

02:27 A squared and b squared were in the equation.

02:30 A score is 21.

02:32 The equation is the parabony, sorry.

02:34 21, b score is 9.

02:37 B score is 12, meaning c is the square root of 12.

02:40 Would use the positive square root...

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