Find the center, foci, and vertices of each ellipse. Graph each equation. 4 x^2+y^2+4 y=0

Name: Problem 53, Chapter 11: Algebra and Trigonometry
Uploaded: 2021-08-17T18:42:23-08:00
Duration: 4 min 6 s
Channel: David Baker
Description: Find the center, foci, and vertices of each ellipse. Graph each equation. 4 x^2+y^2+4 y=0

Find the center, foci, and vertices of each ellipse. Graph...

Key Concepts

Ellipse

An ellipse is a conic section defined as the set of all points in a plane such that the sum of the distances from two fixed points (the foci) is constant. It has a symmetrical shape with a longer dimension called the major axis and a shorter one called the minor axis.

Standard Form of an Ellipse Equation

The standard form of an ellipse's equation, such as (x-h)²/a² + (y-k)²/b² = 1, is essential because it explicitly reveals the ellipse's geometric properties, including its center, the lengths of the semi-axes, and the orientation of the major and minor axes. This form simplifies the process of identifying the ellipse's key features.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic expressions in a way that reveals their structure. For conic sections like ellipses, this technique is vital for transforming a general quadratic equation into its standard form by isolating perfect square trinomials, which in turn makes the center, vertices, and other characteristics of the ellipse easier to determine.

Center, Vertices, and Foci

The center of an ellipse, which is the midpoint of both its major and minor axes, serves as the reference point for its symmetry. The vertices are the points on the ellipse that lie at the extremities of the major axis, indicating the maximum distance from the center along that axis. The foci, located along the major axis at a fixed distance from the center, are fundamental in the definition of an ellipse since the sum of the distances from any point on the ellipse to the two foci is constant.

Transcript

00:01 Here i have an ellipse and i'd like to find the center of fosci and vertices so i can graph it.

00:06 So the first thing that i need to do here is complete the square.

00:11 So 4x squared doesn't have any linear terms, so that doesn't need a square to be completed, but the y does.

00:19 Y squared plus 4 y.

00:20 Here's my 4 magic number, divide it by 2 and square it.

00:25 That gives you 4.

00:26 That's the magic number that will make this a perfect square.

00:29 I just added 4 to the left -hand side of the equation.

00:32 That means i better add 4 to the right -hand side.

00:34 So i did something useful on the left.

00:36 I made all of this a perfect square.

00:39 But i had to add a 4 to do that.

00:41 So i also need to add a 4 on the right -hand side.

00:44 Now i have 4x squared plus y plus 2 squared.

00:49 If i factor that into the perfect square that it is, and i have this equation.

00:54 I'll divide both sides by 4 to make sure that this is in standard form 4 and 11.

01:01 That will look like this standard form because now i have a 1 over here.

01:05 I knew to divide by 4 because i need a 1 on the right.

01:08 This was a 4.

01:11 From here it looks like i have a tall ellipse, not a wide ellipse, because the bigger number is under the y.

01:18 So there's a squared, here's b squared, and we know for an ellipse that a squared is equal to b squared plus c squared that will help us find the fosite.

01:27 So we have 4 equals 1 plus c squared.

01:33 Looks like c is going to be root 3.

01:35 So we've got a is 2, b is 1, and c is root 3...

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