Question

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (4, 0), (4, 14); endpoints of the minor axis: (1, 7), (7, 7)

Answer 1

The standard form of the ellipse is
`((x-4)^(2))/(9) + ((y-7)^(2))/(49) = 1`.

Explanation:

The major axis of the ellipse is located in the y axis, whereas the minor axis is in the x axis. The center of the ellipse is the midpoint of the line segment between vertices, this is:

`(h, k) =(1)/(2)\cdot V_(1) (x,y) + (1)/(2)\cdot V_(2) (x,y)` (1)

If we know that
`V_(1) (x,y) = (4,0)` and
`V_(2)(x,y) = (4, 14)`, then the coordinates of the center are, respectively:

`(h,k) = (1)/(2)\cdot (4, 0) + (1)/(2)\cdot (4,14)`

`(h,k) = (2,0) + (2, 7)`

`(h, k) = (4, 7)`

The length of each semiaxis is, respectively:

`a = \sqrt{(1 - 4)^(2)+(7-7)^(2)}`

`a = 3`

`b = \sqrt{(4-4)^(2)+(0-7)^(2)}`

`b = 7`

The standard equation of the ellipse is described by the following formula:

`((x-h)^(2))/(a^(2))+ ((y-k)^(2))/(b^(2)) = 1`

Where:

`h`,
`k` - Coordinates of the center of the ellipse.

`a`,
`b` - Length of the orthogonal semiaxes.

If we know that
`h = 4`,
`k = 7`,
`a = 3` and
`b = 7`, then the standard form of the ellipse is:

`((x-4)^(2))/(9) + ((y-7)^(2))/(49) = 1`