Real-World Examples: How to Write Ellipse Equations for Orbits and Designs

📚 Quick Study Guide

• 📐 Ellipse Definition: An ellipse is a set of all points such that the sum of the distances to two fixed points (foci) is constant.
• 🔢 Standard Equation (Horizontal Major Axis): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$, where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis.
• 📈 Standard Equation (Vertical Major Axis): $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$, where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis.
• 🎯 Foci: The foci are located at a distance $c$ from the center, where $c^2 = a^2 - b^2$.
• 🛰️ Orbital Application: In orbits, the central body is at one focus of the elliptical path.
• 💡 Design Application: Ellipses are used in architecture and engineering for aesthetic and structural purposes.

Practice Quiz

1. What is the standard form equation of an ellipse centered at (2, -3) with a horizontal major axis of length 10 and a minor axis of length 6?

   1. $\frac{(x-2)^2}{25} + \frac{(y+3)^2}{9} = 1$
   2. $\frac{(x+2)^2}{25} + \frac{(y-3)^2}{9} = 1$
   3. $\frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} = 1$
   4. $\frac{(x+2)^2}{9} + \frac{(y-3)^2}{25} = 1$

2. The orbit of a planet around a star is elliptical. If the star is at one focus, and the major axis has a length of 300 million km, what is the maximum distance the planet can be from the star if the eccentricity is 0.2?

   1. 180 million km
   2. 120 million km
   3. 150 million km
   4. 30 million km

3. An elliptical garden is designed such that its longest length is 16 meters and its shortest width is 10 meters. What is the equation of this ellipse if its center is at the origin and the major axis is along the x-axis?

   1. $\frac{x^2}{64} + \frac{y^2}{25} = 1$
   2. $\frac{x^2}{16} + \frac{y^2}{10} = 1$
   3. $\frac{x^2}{256} + \frac{y^2}{100} = 1$
   4. $\frac{x^2}{8} + \frac{y^2}{5} = 1$

4. What are the coordinates of the foci of the ellipse given by the equation $\frac{(x+1)^2}{16} + \frac{(y-2)^2}{7} = 1$?

   1. (-1, 2 ± 3)
   2. (1, -2 ± 3)
   3. (-1 ± 3, 2)
   4. (1 ± 3, -2)

5. A bridge is designed with a semi-elliptical arch. The arch spans 40 meters and has a maximum height of 12 meters. Assuming the base of the arch is the x-axis and the highest point is on the y-axis, what is the equation of the ellipse?

   1. $\frac{x^2}{400} + \frac{y^2}{144} = 1$
   2. $\frac{x^2}{1600} + \frac{y^2}{144} = 1$
   3. $\frac{x^2}{20} + \frac{y^2}{12} = 1$
   4. $\frac{x^2}{144} + \frac{y^2}{400} = 1$

6. The Earth's orbit around the Sun has an eccentricity of approximately 0.0167. If the semi-major axis is 149.6 million km, what is the approximate distance from the Sun to the other focus of the orbit?

   1. 2.5 million km
   2. 5.0 million km
   3. 1.25 million km
   4. 0.625 million km

7. If the equation of an ellipse is given by $4x^2 + 9y^2 = 36$, find the length of the major axis.

   1. 3
   2. 2
   3. 4
   4. 6