An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b)

An equation of an ellipse is given. (a) Find the center,...

Key Concepts

Major and Minor Axes

The major axis is the longest diameter of the ellipse, running through the center and the vertices, while the minor axis is the shortest diameter, perpendicular to the major axis at the center. The lengths of these axes are twice the lengths of the semi-major and semi-minor axes, respectively.

Foci

The foci are two special points inside the ellipse such that the sum of the distances from any point on the ellipse to each focus is constant. Their positions are determined using the relationship between the semi-major and semi-minor axes, typically through the equation c² = a² - b² (assuming a > b).

Graphing Conic Sections

Sketching an ellipse involves plotting its center, drawing the major and minor axes through the center, and marking vertices and co-vertices along these axes. This provides a framework for accurately drawing the smooth, oval shape of the ellipse, ensuring that key elements like symmetry, vertices, and foci are correctly represented.

Center

The center of an ellipse is the midpoint of the line segment connecting its vertices and is the point around which the ellipse is symmetric. It is determined by the values h and k in the standard equation of the ellipse.

Ellipse

An ellipse is a type of conic section characterized by the set of points for which the sum of the distances from two fixed points (the foci) is constant. Its standard equation, typically written as ((x - h)²)/(a²) + ((y - k)²)/(b²) = 1, reveals important properties such as its center, axes lengths, vertices, and foci.

Vertices

Vertices are the points on the ellipse that lie along the major axis at the maximum distance from the center. They define the extent of the ellipse in the direction of the major axis and are obtained by adding and subtracting the semi-major axis length from the corresponding coordinate of the center.

Transcript

00:01 We have the given equation of ellipses x minus 2 whole square divided by 9 plus y minus 1 whole square divided by 4 equals to 1.

00:16 If we compare this equation with x squared divided by 9 plus y square divided by 4 equals to 1.

00:27 So what here in this case center is origin 0 .0.

00:37 That means now vertex we have, vertex is as it is a horizontal ellipse as 9 is greater than 4, denominator of x squared is greater, so vertex we have plus minus 3 comma 0.

00:53 Forkey, that is equal to square root of 9 minus 4 comma 0 plus minus we can take which is square root of 5 with plus minus comma 0.

01:07 So, it is a shifted ellipse.

01:12 So in this case, we have the center 2 .1.

01:21 Center is 2 .1.

01:23 Now, vertex will try to find.

01:27 So value of a, we know it is 3.

01:31 Value of b, it is 2 squared is 4, so 2.

01:35 Further, if we'll try to find the value of c, so c is square root of a square minus b, so 9 minus 4 is square root of 5.

01:46 Now party of this question, it is being asked what is the center? so we know the center initial in this case it was 0 0.

01:58 So center is sifted by 2 units on xxx and 1 units on y x is in positive direction.

02:03 So center is 0 plus 2, 0 plus 1, which is 2 .1 vertex we have here it was 3 0 minus 3 0.

02:22 So 1 vertex is 3 0 which will become 2 unit on x direction so 3 plus 2 comma 0 plus 1 which is 5 1.

02:39 2 2 2 2 .0 plus 1.

02:47 This is minus 1.

02:52 Next information 4 key we have to find.

02:56 So 4 key here it was root 5 0 minus 0.

02:59 So in this case, foki would be root 5 plus 2 .0 plus 1.

03:05 That is 2 plus root 5 comma 1.

03:10 Next is minus of root 5 plus 2 .0 plus 1, which is 2 minus root 5 comma 1...

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