Exer. 1-14: Find the vertices and foci of the ellipse. Sketch its graph, showing the foci. 4 x^2

Exer. 1-14: Find the vertices and foci of the ellipse....

Key Concepts

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from any point on the ellipse to two fixed points, called foci, is constant. It is a type of conic section that can be oriented horizontally or vertically based on its equation, with its shape determined by the lengths of its major and minor axes.

Completing the Square

Completing the square is an algebraic technique used to convert a quadratic expression into a perfect square plus or minus a constant. In the context of conic sections like ellipses, this method transforms a general second-degree equation into its standard form, making it easier to identify the center, vertices, and orientation of the ellipse.

Standard Form of an Ellipse

The standard form of an ellipse's equation is written as (x - h)²/a² + (y - k)²/b² = 1 for a horizontal ellipse or (x - h)²/b² + (y - k)²/a² = 1 for a vertical ellipse, where (h, k) represents the center, a is the semi-major axis, and b is the semi-minor axis. This form simplifies the process of analyzing the ellipse’s geometric properties.

Vertices of an Ellipse

The vertices of an ellipse are the points where it intersects its major axis. They lie at a distance of a (the semi-major axis) from the center along the direction of the major axis, serving as the furthest points from the center on the ellipse.

Foci of an Ellipse

The foci are specific points inside the ellipse that play a key role in its definition; the sum of the distances from any point on the ellipse to the two foci is constant. They are located along the major axis and can be found using the relationship c² = a² - b², where c is the distance from the center to each focus.

Transcript

00:01 The first thing i would do with this problem is rearrange the order and put my x's together.

00:08 It's 32.

00:09 I like to put a space right there and then deal with the y's.

00:14 I don't really need the space yet, but i will minus 36y is equal to, i'm going to subtract that 64 over and 0 minus 64 is negative 64.

00:30 So the first thing i want to do is factor out at 4.

00:34 Because i want this leading coefficient to be one, and then just stop right there.

00:40 And same thing with the y is i need to factor out of 9, and 36 divide my 9 is 4.

00:51 And what i need to do is think about what numbers need to go in there to complete the square.

00:57 And the process is always taking that b value divided by 2 squared, and negative 4 squared is 16.

01:04 But what i'm really adding right here is the quantity of 4 times 16.

01:10 So the left side of the equation is changing by 64 units.

01:15 So i need to also add the right side to 64.

01:19 And the same thing goes with this middle piece, this blue piece divided by 2 squared equals 4.

01:28 But what i'm really adding to it is 36.

01:31 9 times 4 is 36.

01:34 So as i start to clean this up, first of all, we start with four, and then the red piece is now factorable to be x minus four, x minus four, so i can write it once squared.

01:47 Same thing with what's in blue.

01:50 It factors to be y minus two, y minus two, i can just write it once squared.

01:55 And on the right side, well, those 64s are canceled, and i'm left with 36.

01:59 But it's still not in the form i want.

02:01 I want it to equal one.

02:03 I can fix that by dividing by 36.

02:06 So i also have to divide this by 36 and divide this by 36...