# Worksheet: Equation of an Ellipse

In this worksheet, we will practice finding the equation of an ellipse using different givens and using it to solve problems that involve ellipse-shaped constructions.

**Q2: **

An ellipse has a vertex at and a covertex at . The major axis of the ellipse is parallel to the -axis. What is the equation of the ellipse?

- A
- B
- C
- D
- E

**Q3: **

Derive the equation of an ellipse centered at the origin with foci at the points and , which has a major axis of length 10.

- A
- B
- C
- D
- E

**Q6: **

An arch that is 12 feet high and 40 feet wide is shaped like the top half of an ellipse. Determine the equation of this ellipse and, using it, find the distance from its center to the point where the height of the arch is 6 feet. Give your answer to the nearest hundredth of a foot.

- A, distance = 17.32 ft
- B, distance = 19.36 ft
- C, distance = 30 ft
- D, distance = 300 ft
- E, distance = 5.48 ft

**Q7: **

An arch that is 8 feet high and 20 feet wide is shaped like the top half of an ellipse. Determine the equation of this ellipse, and use it to find the height of the arch at a distance of 4 feet from its center. Give your answer to the nearest hundredth.

- A, height = 53.76 ft
- B, height = 61.44 ft
- C, height = 7.33 ft
- D, height = 1.79 ft
- E, height = 7.84 ft

**Q8: **

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

- A ft
- B ft
- C ft
- D90 ft
- E30 ft

**Q9: **

Derive the equation of an ellipse centered at the origin with foci at the points and , which has its vertices at the points and .

- A
- B
- C
- D
- E

**Q11: **

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus and the other focus is 80 feet away, what is the length and height at the center of the gallery?

- Alength = 88 ft, height = ft
- Blength = 88 ft, height = ft
- Clength = 80 ft, height = ft
- Dlength = 96 ft, height = ft
- Elength = 96 ft, height = ft