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.

how can i solve this please

Answer 1

Answer on Question #46884 – Math – Analytic 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.

Standard form of equation for an ellipse with horizontal major axis:

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1, \text{ where } a > b, (h, k) = (x, y) \text{ coordinates of center}$$

For given ellipse, center: $(0, 0)$

length of horizontal major axis: $10 = 2a \rightarrow a = 5$

length of minor axis: $4 = 2b \rightarrow b = 2$

Thus, equation for given ellipse:

$$\frac{x^2}{25} + \frac{y^2}{4} = 1.$$

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