Find the center, foci, and vertices of each ellipse. Graph each equation. x^2+4 x+4 y^2-8 y+4=0

Name: Problem 47, Chapter 11: Algebra and Trigonometry
Uploaded: 2021-08-17T18:27:53-08:00
Duration: 5 min 38 s
Channel: David Baker
Description: Find the center, foci, and vertices of each ellipse. Graph each equation. x^2+4 x+4 y^2-8 y+4=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 for which the sum of the distances to two fixed points (foci) is constant. Its standard form equation usually involves squared terms for x and y, and it is characterized by a center, vertices, and foci, which together reveal its shape and orientation.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic expressions in a perfect square form. This technique is crucial for converting the general quadratic equation of a conic section into its standard form, which then makes it easier to identify features like the center, vertices, and foci.

Center

The center of an ellipse is the point that lies midway between its vertices and foci. In the standard form of an ellipse's equation, the center (h, k) indicates the translation from the origin, and it is the point about which the ellipse is symmetric.

Vertices

Vertices are the points on the ellipse that lie along its major axis and are the farthest from the center. They determine the maximum and minimum extents of the ellipse along its principal direction and are directly used to measure the length of the major axis.

Foci

The foci (plural of focus) are two specific points inside the ellipse that have the property that the sum of the distances from any point on the ellipse to the foci is constant. The distance between the center and each focus is related to the ellipse's eccentricity and helps determine the shape of the ellipse.

Graphing Conic Sections

Graphing conic sections involves plotting the key features such as the center, vertices, and foci, and drawing the curve based on these points. For an ellipse, understanding its standard form and how to compute these elements is essential to accurately sketch its shape and orientation.

Transcript

00:01 Okay, here we have an ellipse and we need to complete the square so we can find the center, the vertices, and the foci.

00:07 When you complete the square in a quadratic, quadratic will look like a x squared plus bx plus c, we need to focus on the b.

00:17 The magic number that we're going to use is half of b squared.

00:21 So keep that in mind as we go through this.

00:25 So the first thing i want to do is change this.

00:28 I'm going to factor out on the y.

00:30 I want to have that y squared by itself, so i'm going to factor out of four.

00:34 Get this y squared by itself.

00:39 And i'll move this four to the other side.

00:43 So now i have two quadratics, and i'm ready to complete the square.

00:47 So on the first one, here's b.

00:51 Let's take half of it in squared.

00:53 Half of four is two.

00:54 Two squared is coincidentally back to four.

00:57 On the second quadratic, keep the four factored out.

01:02 Here's the b i want, negative 2.

01:05 Half of negative 2 is negative 1.

01:07 Negative 1 squared is 1.

01:09 So i'm going to add 1 here.

01:12 Now, since i added this 4, and i put this 1 here, i can't just add those to the left.

01:21 It's useful because it's going to help me complete the square, but i have to also add them to the right.

01:25 So i'm going to add 4.

01:28 That's from this one.

01:30 And then this 1, it looks like i should add 1 to the other side, but i have to remember that that one is actually multiplying by this 4.

01:38 So i have to add 4 again to the other side.

01:41 Now i can factor these.

01:43 Now they're all going to be perfect squared.

01:47 X plus 2 squared plus 4 times y minus 1 squared equals these cancel.

01:53 And i have 4 over here.

01:55 And now i simply want to have a 1 here instead of a 4.

01:59 So i'm going to divide everything by 4 to put this in standard form for an ellipse.

02:03 So this gets divided by four, the second term, the four will cancel when you divide by four, and it'll just be over one, and this equals one here.

02:14 And now i have the equation of the ellipse...

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