Analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each

Analyze each equation; that is, find the center, foci, and...

Key Concepts

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 (foci) is constant. It is characterized by its center, vertices, and foci, which together determine its overall shape and orientation.

Standard Form of an Ellipse

The standard form of an ellipse is expressed as (x-h)²/a² + (y-k)²/b² = 1, where (h,k) represents the center of the ellipse, and a and b denote the semi-major and semi-minor axes respectively. Converting an ellipse's equation into this form is crucial for easily identifying its key features such as the center, vertices, and foci.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a perfect square format. This process is essential for transforming a quadratic equation from its general form into the standard form of a conic section like an ellipse, making the geometric properties more apparent.

Center

The center of an ellipse is the midpoint of its major and minor axes, and it represents the point from which the ellipse is symmetrically distributed. This point is readily identifiable once the ellipse's equation is rewritten in standard form.

Vertices

Vertices are the points where the ellipse intersects its major axis, marking the maximum and minimum extents along that axis. The distance from the center to a vertex is equal to the length of the semi-major axis, and these points are key in defining the ellipse's shape.

Foci

The foci are two fixed points located along the major axis inside the ellipse. They are fundamental to the ellipse's definition because the sum of the distances from any point on the ellipse to the foci is constant. The distance from the center to each focus is determined by the relationship c² = a² - b², where a and b are the semi-major and semi-minor axes respectively.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci, and then drawing a smooth, closed curve that conforms to the ellipse's geometric definitions. The process requires scaling the coordinate axes appropriately based on the lengths of the semi-major and semi-minor axes to accurately represent the ellipse's proportions.

Transcript

00:01 So we are given this equation, which is x squared plus 4x plus 4y squared minus 8y plus 4.

00:16 So the first thing we have to do is set this up so that we can analyze it.

00:22 So we are going to have x squared plus 4x plus 4y squared minus 8y equal to negative 4.

00:35 Since we moved that over to this side.

00:39 And we are going to divide all of this by negative 4 so that we have 1 on this side.

00:45 So now we have x squared, or 1 4th, x squared.

00:52 Actually, we're not going to do that just yet.

00:54 So we have x squared plus 4x plus 4 times y squared minus 2y.

01:05 It's equal to negative 4.

01:07 So what we're going to have to do to put this in the equation that we want, which is x minus a squared over a squared, plus y minus k squared over b squared equals 1, is to complete the square.

01:27 So on this side, we're going to have to do 1 half b squared.

01:32 So that's going to be 2 squared, which is 4.

01:36 And inside the parentheses we have to do 4 minus 4 times y squared minus 2 y so one half of negative 2 is negative 1 squared is 1 so that's going to give us 1 over here and that's going to be equal to negative 4 and since we added a 4 over here we're going to have to add a 4 here since we added 1 here we have to multiply it by this 4 and then add it here and then add it here as well.

02:09 So we have another plus 4.

02:11 So now we have, and since we're completing the square, we can factor this and we'll get x plus 2 squared plus 4 times y minus 1 squared equal to 8.

02:36 So now we have to divide through by 8, which will give us x plus 2 squared over 8.

02:46 Plus y minus 1 squared over 2 equal to 0.

02:53 Not 0.

02:54 What am i saying? we divided that by 8.

02:57 So that's going to give us 1.

02:59 Now we have a format that we can work with.

03:01 This matches the equation we have up here.

03:04 So i'm going to clear this out so we can conduct our analysis.

03:16 And i'm going to move this guy here.

03:30 So now what we're going to do is we're going to analyze this.

03:33 So you can tell that 8 is greater than 2, which means that this ellipse has a major axis parallel to the x -axis.

03:41 So we are using the equation x plus or x minus h squared over a squared plus y minus k squared over b squared equal to 1.