Find the vertices and foci of the ellipse and sketch its...

Key Concepts

Foci

Foci are two fixed points inside the ellipse that have the property that the sum of the distances from any point on the ellipse to both foci is constant. The location of the foci relative to the center depends on the distances between the axes (a and b), and finding them is crucial for both the geometric definition of the ellipse and its application in various fields.

Vertices

Vertices are the points where the ellipse intersects its major axis. They represent the extremal points — the farthest distances from the center along the principal axes. Determining the vertices from the standard form is key to sketching the ellipse and understanding its overall shape and dimensions.

Ellipse

An ellipse is a type of conic section that can be defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. This concept is fundamental in understanding how quadratic equations represent curved shapes and is widely applicable in physics, astronomy, and engineering.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a form that makes its properties more apparent. In the context of conic sections like ellipses, it’s used to convert the general quadratic form into the standard form of an ellipse, revealing the center and allowing easy determination of other key properties.

Standard Form of an Ellipse

The standard form of an ellipse, typically written as ((x-h)²/a²) + ((y-k)²/b²) = 1, clearly displays the center (h, k) and the lengths of the semi-major and semi-minor axes (a and b). This form is essential for analyzing the geometric characteristics of the ellipse and for performing further calculations to find vertices, foci, and for graphing.

Transcript

00:01 To put this in standard form into graph it.

00:02 So the first thing you need to do is to pull a 25 out of these first two terms and put that out front right there.

00:10 That leaves you with the y squared plus 2y.

00:13 Now we need to complete the square on the inside.

00:15 Remember in order to complete the square, you add b over 2 squared.

00:19 So the b here is negative 2.

00:20 So i added negative 2 over 2 squared.

00:22 So if i add that here, i'm going to have to add it on the other side too or i change the equation.

00:27 So i need to add the...

00:29 Notice that i'm actually distributing the 25 times that value.

00:33 So over here i need to add 25 times the negative 2 over 2 squared, or i change the equation.

00:41 So now i'm going to just go 4x squared plus 25 times.

00:46 Now you're going to factor this.

00:48 Negative 2 divided by 2 squared is actually just 1.

00:51 So factors of 1 that add to negative 2 be negative 1 and negative 1, so it's y minus 1 squared, equals and you've got 75 plus 25 which of course is just going to give me the 100 so i've got 100 here okay to get it in standard form i need to divide by a hundred in order to graph it all right because it's got to be equal to 1 so 4 goes into 125 times and then y minus 1 25 goes into 104 times equals 1 okay, from the equation, i can see that the center is going to be 0 comma 1.

01:32 So here's my center right here.

01:35 A is 5 and it's underneath the x.

01:38 Remember, it's the biggest one.

01:39 So i'm going to go sideways 5.

01:41 1, 2, 3, 4, 5.

01:43 So there's a vertex and this way 5, 1, 2, 3, 4, 5...

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