Sketching an Ellipse In Exercises 25-30 , find the center, foci, vertices, and eccentricity of t

Sketching an Ellipse In Exercises 25-30 , find the center,...

Key Concepts

Ellipse

An ellipse is a type of conic section that results from the intersection of a plane with a right circular cone, where the plane cuts through both halves of the cone. It is 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 to understanding the geometric properties and equations associated with ellipses.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a standard form. In the context of ellipses, this process allows one to rewrite a general quadratic equation into the standard form of an ellipse, from which key information such as the center, axes lengths, and orientation can be determined.

Center of the Ellipse

The center of an ellipse is the midpoint of the line segment joining its vertices along each axis, and it represents the point around which the ellipse is symmetric. Identifying the center is essential as it serves as the reference point for locating other features such as the vertices and foci.

Vertices and Axes

Vertices are the endpoints of the major and minor axes of an ellipse and represent the farthest points from the center along these axes. Determining the positions of the vertices and lengths of the axes is crucial for understanding the size and orientation of the ellipse.

Foci

The foci are two special points located along the major axis of an ellipse. They have the defining property of the ellipse: the sum of the distances from any point on the ellipse to each focus is constant. The location of the foci relative to the center influences the shape of the ellipse.

Eccentricity

The eccentricity of an ellipse is a measure of its deviation from being circular. It is defined as the ratio of the distance from the center to a focus over the semi-major axis length. An eccentricity of 0 corresponds to a circle, while values closer to 1 indicate a more elongated ellipse.

Transcript

00:01 Hi guys, this problem is to ask about the ellipse with the equation 9x square plus 4 y square plus 36x minus 24y minus 36 equals to 0.

00:18 So the first thing is we must convert this equation to the standard ellipse equation.

00:25 So we need to complete square for this equation.

00:35 So we put all the term with x together.

00:46 So it's 9x squared plus 36x plus 4 y squared minus 24 y and then move the 36 to the other side.

01:08 And then we can factor the 9 out for x and factor 4 out for y.

01:14 So it's 9 x squared plus 4x plus 4x plus 4.

01:21 Y square minus 6y equals to 36 okay in order to complete square we have to plus 4 for the x and plus 9 for the y so it's 9 x squared plus 4x plus 4 but we add 4 we must minus the 9 times 4 to balance this equation and then plus 4 y squared minus 6y plus 9 and then minus 4 times 9 in order to balance this equation equals to 36.

02:24 So we can complete square for this equation, that is 9 x plus 2 square minus 36 plus 4y minus 3 square minus 36 equals to 36.

02:52 Okay, so now we divide, we move both 36 to the other side.

03:03 We get 108 on the other side, and then we divide by 108 on the both sides to convert it to a standard equation of an ellipse.

03:25 So that is 27.

03:30 Okay, so we see the denominator below the y square is greater than the denominator below the x square.

03:42 So it's still an ellipse like this shape.

03:53 Okay, so the a is a square is 27, b square is 12, so the c square will be a square minus b square that is 15.

04:08 So the a is square root of 27, so it's 3, square root 3...

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