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

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci, and then drawing a smooth, symmetric curve that passes through the vertices. This process provides a visual representation of the ellipse's geometric properties, such as its orientation, dimensions, and the relationship between its axes.

Major and Minor Axes

The major axis is the longest line segment that passes through the center of the ellipse, with its endpoints being the vertices. Its length is 2a. The minor axis is the perpendicular segment through the center, with a length of 2b. These axes define the overall dimensions of the ellipse.

Foci

The foci (plural of focus) are two special points located along the major axis of the ellipse from which the distance to any point on the ellipse has a constant sum. Their distance from the center is calculated using the relationship c² = a² - b², where c is the distance from the center to each focus.

Vertices

Vertices are the points on the ellipse that lie at the endpoints of the major axis and represent the farthest distances from the center in that direction. Their locations are determined by adding and subtracting the value of the semi-major axis from the appropriate coordinate of the center.

Center of an Ellipse

The center of an ellipse is the point from which the distances to the vertices and foci are measured. In the standard equation of an ellipse, the center is given by the coordinates (h, k), which also serve as the midpoint of both the major and minor axes.

Standard Form of an Ellipse

The standard form of an ellipse is written as ((x - h)² / a²) + ((y - k)² / b²) = 1, where (h, k) is the center of the ellipse. The parameters a and b represent the distances from the center to the ellipse along the x and y directions, respectively, and they determine the shape and orientation of the ellipse.

Transcript

00:01 Equation x plus 1 whole square divide by 36 plus y plus 1 whole square it is y plus 1 whole square divided by 64 is 1 so if we compare this equation with a standard ellipse x square divide by 36 plus y square divided by 64 equals to 1 and corresponding ellipse shifted ellipse we have x plus 1 whole square divide by 36 plus y square y minus 1 y plus 1 whole square divided by 64 equals to 1 in this case we are getting the value of a is it is 8 a square is 6 value of b is 6 corresponding value of c will get square root of 64 minus 36 which is 64 minus 36 we have 36 64.

01:19 So 64 minus 36 we should write 28.

01:23 So 2 root 7.

01:27 So here the center we have 0 0 vertex we have it is vertical ellipse so 08 and and 0 minus 8.

01:48 Ok, we have 0 plus minus c, so 0 2 root 7 and 0 negative 2 root 7.

02:02 Now, point b part of this question, length of major axis, which is equal to 2 times 8, that is 16 units, and length of minor axis, which is 2.

02:31 Times b that is equal to b value we have 6 so 2 times 6 is 12 units now the same value will calculate for this so this is center is sifted by minus 1 minus 1 unit so in this case part 8 center we have as this ellipse is sifted by minus 1 minus 1 units so we should write it is sifted by minus 1 comma minus 1 units in x and y direction respectively so center is 0 0 so 0 it is minus 1 minus 1 vertex will have it is 0 minus 1 that is minus 1 and 8 minus 1 is 7 and 0 minus 1 and minus 1 and minus 8 minus 9 4 we have 0 minus 1 is minus 1 and 2 root 7 minus 1 and next is minus 1 and minus 1 and next is minus 1 minus 2 root 7 minus 1 length of major and minor axis remains same length of major axis is 2 times 8 at the same as 16 units length of minor axis is 2 times 6 is 12 units now it will represent this ellipse on coordinate axis so it is minus 1 minus 1 septal center is minus 1 minus 1 so minus 1 7 7 minus 1 9 so let's say it is y axis and it is minus 1 minus 9 minus 7 so on y axis it is vertex so minus 9 we should have 9 7 16 units so 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 so kind of of it is a vertical ellipse so we should draw the line it is x -axis it is minus 1 minus 9 is the vertex so it is minus 1 minus 9 minus 1 minus 1 minus 2 minus 3 minus 4 minus 5 minus 6 minus 6 so let's say it is minus 1 minus 9 and minus 1 7 we have this point in x direction we have to sift it right it was 6 so so we should go write 6 units...

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