Question

An elliptical riding path is to be built on a rectangular piece of property as shown below.

The rectangular piece of property measures 10 mi by 4 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

Answer on Question #45831 – Math – Analytical Geometry

An elliptical riding path is to be built on a rectangular piece of property as shown below.

The rectangular piece of property measures 10 mi by 4 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 sides.

Solution:

General Equation of an Ellipse: an ellipse can be defined as the locus of all points that satisfy the equation:

$$\frac {x ^ {2}}{a ^ {2}} + \frac {y ^ {2}}{b ^ {2}} = 1$$

where $x, y$ are the coordinates of any point on the ellipse, $(a, 0), (-a, 0)$ and $(0, b), (0, -b)$ are the vertices on the $x$ and $y$ axes respectively. Here $10 = 2a$, $4 = 2b$, hence $a = 5$, $b = 2$.

In our case:

$$\frac {x ^ {2}}{5 ^ {2}} + \frac {y ^ {2}}{2 ^ {2}} = 1$$

Answer: equation of the ellipse is

$$\frac {x ^ {2}}{2 5} + \frac {y ^ {2}}{4} = 1$$

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