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Worksheet: Equations of Ellipses

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 π‘₯ βˆ’ 7 ) 5 0 + ( 𝑦 βˆ’ 3 ) 2 2 5 = 1 2 2
  • B ( 2 π‘₯ βˆ’ 3 ) 2 2 5 + ( 𝑦 βˆ’ 7 ) 5 0 = 1 2 2
  • C ( 2 π‘₯ βˆ’ 3 ) 5 0 + ( 𝑦 βˆ’ 7 ) 2 2 5 = 1 2 2
  • D ( 2 π‘₯ βˆ’ 7 ) 2 2 5 + ( 𝑦 βˆ’ 3 ) 5 0 = 1 2 2
  • E ( π‘₯ βˆ’ 7 ) 2 2 5 + ( 𝑦 βˆ’ 3 ) 5 0 = 1 2 2

Q2:

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

  • A π‘₯ 8 + 𝑦 4 = 1 2 2
  • B π‘₯ 6 4 + 𝑦 1 6 = 1 2 2
  • C π‘₯ 4 + 𝑦 2 = 1 2 2
  • D π‘₯ 1 6 + 𝑦 4 = 1 2 2
  • E π‘₯ 1 6 + 𝑦 8 = 1 2 2

Q3:

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

  • A π‘₯ 1 6 + 𝑦 2 5 = 1  
  • B π‘₯ 9 + 𝑦 2 5 = 1  
  • C π‘₯ 2 5 + 𝑦 1 6 = 1  
  • D π‘₯ 2 5 + 𝑦 9 = 1  
  • E π‘₯ 5 + 𝑦 3 = 1  

Q4:

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 π‘₯ 4 0 0 + 𝑦 6 4 = 1 2 2 , height = 61.44 ft
  • B π‘₯ 1 0 0 + 𝑦 6 4 = 1 2 2 , height = 53.76 ft
  • C π‘₯ 4 0 0 + 𝑦 6 4 = 1 2 2 , height = 7.84 ft
  • D π‘₯ 1 0 0 + 𝑦 6 4 = 1 2 2 , height = 7.33 ft
  • E π‘₯ 2 0 + 𝑦 1 6 = 1 2 2 , height = 1.79 ft

Q5:

An ellipse has a vertex at ( 1 1 , 5 ) and a covertex at ( 6 , 8 ) . The major axis of the ellipse is parallel to the π‘₯ -axis. What is the equation of the ellipse?

  • A π‘₯ 9 + 𝑦 2 5 = 1 2 2
  • B π‘₯ 2 5 + 𝑦 9 = 1 2 2
  • C ( π‘₯ βˆ’ 6 ) 5 + ( 𝑦 βˆ’ 5 ) 3 = 1 2 2
  • D ( π‘₯ βˆ’ 6 ) 2 5 + ( 𝑦 βˆ’ 5 ) 9 = 1 2 2
  • E ( π‘₯ βˆ’ 5 ) 2 5 + ( 𝑦 βˆ’ 6 ) 9 = 1 2 2

Q6:

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high, denoting the height by β„Ž .

  • A π‘₯ 2 β„Ž + 𝑦 0 . 5 β„Ž = 1 2 2 2 2
  • B π‘₯ 1 6 β„Ž + 𝑦 β„Ž = 1 2 2 2 2
  • C π‘₯ 4 β„Ž + 𝑦 β„Ž = 1 2 2 2 2
  • D π‘₯ 4 β„Ž + 𝑦 0 . 2 5 β„Ž = 1 2 2 2 2
  • E π‘₯ 2 β„Ž + 𝑦 β„Ž = 1 2 2 2 2

Q7:

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 90 ft
  • B 30 ft
  • C 6 0 √ 3 ft
  • D 3 0 √ 3 ft
  • E 3 0 √ 1 5 ft

Q8:

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

  • A π‘₯ 4 8 + 𝑦 1 6 = 1 2 2
  • B π‘₯ 4 8 + 𝑦 6 4 = 1 2 2
  • C π‘₯ 1 6 + 𝑦 6 4 = 1 2 2
  • D π‘₯ 6 4 + 𝑦 6 0 = 1 2 2
  • E π‘₯ 2 5 + 𝑦 6 4 = 1 2 2

Q9:

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 π‘₯ 4 0 0 + 𝑦 5 7 6 = 1 2 2 , distance = 19.36 ft
  • B π‘₯ 4 0 0 + 𝑦 1 4 4 = 1 2 2 , distance = 300 ft
  • C π‘₯ 4 0 + 𝑦 1 4 4 = 1 2 2 , distance = 30 ft
  • D π‘₯ 4 0 0 + 𝑦 1 4 4 = 1 2 2 , distance = 17.32 ft
  • E π‘₯ 4 0 + 𝑦 1 4 4 = 1 2 2 , distance = 5.48 ft