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.

Q1:

Derive the equation of an ellipse with foci at the points (1,3) and (6,3), which has a major axis of length 15.

  • A ( 2 𝑥 3 ) 2 2 5 + ( 𝑦 7 ) 5 0 = 1
  • B ( 2 𝑥 7 ) 2 2 5 + ( 𝑦 3 ) 5 0 = 1
  • C ( 2 𝑥 7 ) 5 0 + ( 𝑦 3 ) 2 2 5 = 1
  • D ( 𝑥 7 ) 2 2 5 + ( 𝑦 3 ) 5 0 = 1
  • E ( 2 𝑥 3 ) 5 0 + ( 𝑦 7 ) 2 2 5 = 1

Q2:

An ellipse has a vertex at (11,5) and a covertex at (6,8). The major axis of the ellipse is parallel to the 𝑥-axis. What is the equation of the ellipse?

  • A 𝑥 2 5 + 𝑦 9 = 1
  • B ( 𝑥 6 ) 2 5 + ( 𝑦 5 ) 9 = 1
  • C ( 𝑥 6 ) 5 + ( 𝑦 5 ) 3 = 1
  • D 𝑥 9 + 𝑦 2 5 = 1
  • E ( 𝑥 5 ) 2 5 + ( 𝑦 6 ) 9 = 1

Q3:

Derive the equation of an ellipse centered at the origin with foci at the points (4,0) and (4,0), which has a major axis of length 10.

  • A 𝑥 5 + 𝑦 3 = 1
  • B 𝑥 2 5 + 𝑦 9 = 1
  • C 𝑥 9 + 𝑦 2 5 = 1
  • D 𝑥 1 6 + 𝑦 2 5 = 1
  • E 𝑥 2 5 + 𝑦 1 6 = 1

Q4:

Find the equation of the ellipse that fits just inside a box that is 8 units high and 4 units wide.

  • A 𝑥 6 4 + 𝑦 1 6 = 1
  • B 𝑥 8 + 𝑦 4 = 1
  • C 𝑥 4 + 𝑦 2 = 1
  • D 𝑥 1 6 + 𝑦 4 = 1
  • E 𝑥 1 6 + 𝑦 8 = 1

Q5:

Find the equation of an ellipse in standard position whose width is four times its height. Let the height of the ellipse be .

  • A 𝑥 1 6 + 𝑦 = 1
  • B 𝑥 2 + 𝑦 0 . 5 = 1
  • C 𝑥 4 + 𝑦 0 . 2 5 = 1
  • D 𝑥 4 + 𝑦 = 1
  • E 𝑥 2 + 𝑦 = 1

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 𝑥 4 0 0 + 𝑦 1 4 4 = 1 , distance = 17.32 ft
  • B 𝑥 4 0 0 + 𝑦 5 7 6 = 1 , distance = 19.36 ft
  • C 𝑥 4 0 + 𝑦 1 4 4 = 1 , distance = 30 ft
  • D 𝑥 4 0 0 + 𝑦 1 4 4 = 1 , distance = 300 ft
  • E 𝑥 4 0 + 𝑦 1 4 4 = 1 , 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 𝑥 1 0 0 + 𝑦 6 4 = 1 , height = 53.76 ft
  • B 𝑥 4 0 0 + 𝑦 6 4 = 1 , height = 61.44 ft
  • C 𝑥 1 0 0 + 𝑦 6 4 = 1 , height = 7.33 ft
  • D 𝑥 2 0 + 𝑦 1 6 = 1 , height = 1.79 ft
  • E 𝑥 4 0 0 + 𝑦 6 4 = 1 , 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 3 0 1 5 ft
  • B 6 0 3 ft
  • C 3 0 3 ft
  • D90 ft
  • E30 ft

Q9:

Derive the equation of an ellipse centered at the origin with foci at the points (2,0) and (2,0), which has its vertices at the points (8,0) and (8,0).

  • A 𝑥 1 6 + 𝑦 6 4 = 1
  • B 𝑥 4 8 + 𝑦 1 6 = 1
  • C 𝑥 2 5 + 𝑦 6 4 = 1
  • D 𝑥 6 4 + 𝑦 6 0 = 1
  • E 𝑥 4 8 + 𝑦 6 4 = 1

Q10:

A bridge is to be built in the shape of a semielliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

  • A 2 5 6 9 ft
  • B 2 4 5 5 ft
  • C 5 7 6 5 ft
  • D 1 6 3 ft
  • E 1 2 1 4 7 ft

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 = 821 ft
  • Blength = 88 ft, height = 811 ft
  • Clength = 80 ft, height = 819 ft
  • Dlength = 96 ft, height = 885 ft
  • Elength = 96 ft, height = 811 ft

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