Question

The rectangular piece of property measures 8 mi by 6 mi. find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides

Answer 1

The ellipse will be inscribed in the rectangle.
Because the rectangle measures 8 mi by 6 mi, the ellipse has
a = 8/2 = 4 mi (major axis)
b = 6/2 = 3 mi (minor axis)

The equation for the ellipse is
x^2/a^2 + y^2/b^2 = 1

That is
x^2/4^2 + y^2/3^2 = 1
x^2/16 + y^2/9 = 1

Answer:
x^2/16 + y^2/9 = 1

Answer 2

The equation of ellipse is
`(x^2)/(9)+(y^2)/(16)=1`.

Explanation:

Consider the provided information.

The rectangular piece of property measures 8 mi by 6 mi.

It is a vertical ellipse.

So, use the equation:
`(x^2)/(b^2)+(y^2)/(a^2)=1`

Where the value of a and b are:

`a=(8)/(2)=4\\b=(6)/(2)=3`

The major axis is along y-axis with the vertices (0,4) and (0,-4).

The minor axis is along the x-axis, with the vertices (-3,0) and (3,0).

Now substitute the value of a and b in
`(x^2)/(b^2)+(y^2)/(a^2)=1`

`(x^2)/(3^2)+(y^2)/(4^2)=1`

`(x^2)/(9)+(y^2)/(16)=1`

Hence, the equation of ellipse is
`(x^2)/(9)+(y^2)/(16)=1`.