Find the standard form of the equation of an ellipse with the given characteristics. Vertices (-

step 1 We want to write the equation of the ellipse with our vertices. I'm just going to write V for ve

Find the standard form of the equation of an ellipse with...

Key Concepts

Standard Form of an Ellipse Equation

The standard form of an ellipse's equation is a way to express the ellipse in a concise algebraic format, typically written as ((x - h)²)/(a²) + ((y - k)²)/(b²) = 1, where (h, k) represents the center of the ellipse and a and b represent the distances from the center to the ellipse along the horizontal and vertical axes (or vice versa), respectively. This form clearly separates the contributions of the x and y components, making it easy to analyze the geometric properties of the ellipse.

Center of the Ellipse

The center of an ellipse is the point that is equidistant from all pairs of corresponding points along the ellipse's axes. It is usually found as the midpoint of the line segment that connects the vertices or the midpoints of the endpoints of both the major and minor axes. The center plays a key role in determining the position of the ellipse in the coordinate plane.

Vertices and Co-vertices

Vertices are the endpoints of the ellipse along the major axis and represent the farthest points from the center along that axis. Co-vertices, on the other hand, are the endpoints along the minor axis. Together, these points determine the shape and dimensions (lengths of the major and minor axes) of the ellipse, and they are integral to formulating the standard equation.

Major and Minor Axes

The major axis of an ellipse is the longest diameter that passes through the center and connects two vertices, while the minor axis is the shortest diameter that also passes through the center and connects the co-vertices. The lengths of these axes (determined by 2a and 2b, respectively) are crucial for understanding the size and orientation of the ellipse.

Orientation of the Ellipse

The orientation of an ellipse refers to its alignment in the coordinate plane. It is determined by the direction of the major axis, which can either be horizontal or vertical. This orientation affects how the equation of the ellipse is written in standard form, as the roles of a and b in the equation will switch depending on whether the ellipse is stretched more along the x-axis or the y-axis.

Transcript

00:01 We want to write the equation of the ellipse with our vertices.

00:04 I'm just going to write v for vertices.

00:07 Those vertices are at negative 1, negative 9, and negative 1 -1.

00:21 Let's plot those two points.

00:25 So there's negative 1, and then 1, 2, 3, 4, 5, 6, 7, 8, 9.

00:34 So there's negative 1, negative 9, and negative 1, and negative 1, then the endpoints of the minor axis are negative 4 and negative 4 and then the other one is 2 negative 4 so 1 2 3 4 1 2 3 4 there's 1 and then 2 negative 4 will be here so that ellipse will look like this, notice that our center is the midpoint of the vertices.

01:23 It's also the midpoint of the endpoints of the minor axis.

01:28 And so our center here, when you add negative 4 plus 2, negative 2 divided by 2 is negative 1.

01:35 And if you look at the vertices, negative 1 plus negative 1 is negative 2 divided by 2 gives you negative 1.

01:42 Now let's get the y coordinate of the center.

01:44 Negative 9 plus 1 is negative 8 divided by 2 is negative 4 and if you look at the end points of the minor axis negative 4 plus negative 4 is negative 8 divided by 2 would be negative 4 so there's our center now we've got to find our a and b and a we usually let be the larger so for a we're going to count from the center to a point on the ellipse vertically.

02:16 So how many units do we have to count up and down? and it looks like five units.

02:25 Now our b will be how many units we have to count from the center horizontally to a point on the ellipse...

Please give Ace some feedback