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Lesson: Equations of Hyperbolas

Sample Question Videos

Worksheet • 10 Questions • 1 Video

Q1:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the center of a park. The hedge will follow the asymptotes 𝑦 = 2 3 π‘₯ and 𝑦 = βˆ’ 2 3 π‘₯ , and its closest distance to the center fountain is 12 yards. Find the equation of the hyperbola.

  • A π‘₯ 1 4 4 βˆ’ 𝑦 6 4 = 1 2 2
  • B π‘₯ 1 4 4 βˆ’ 𝑦 3 2 4 = 1 2 2
  • C π‘₯ 1 2 βˆ’ 𝑦 8 = 1 2 2
  • D π‘₯ 1 4 4 βˆ’ 𝑦 8 = 1 2 2
  • E π‘₯ 2 4 βˆ’ 𝑦 8 = 1 2 2

Q2:

Suppose that we model an object’s trajectory in the solar system by a hyperbolic path in the coordinate plane. The π‘₯ -axis is a line of symmetry of this hyperbola. The object enters in the direction of 𝑦 = 0 . 5 π‘₯ + 2 and leaves in the direction 𝑦 = βˆ’ 0 . 5 π‘₯ βˆ’ 2 . The sun is positioned at the origin and the object passes within 1 AU (astronomical unit) of the sun at its closest. Using the asymptote’s equations, find the equation of the object’s path.

  • A ( π‘₯ + 4 ) 9 βˆ’ 4 𝑦 9 = 1 2 2
  • B ( π‘₯ + 4 ) 3 βˆ’ 𝑦 7 = 1 2 2
  • C ( π‘₯ + 4 ) βˆ’ 𝑦 1 6 = 1 2 9 4 2
  • D ( π‘₯ βˆ’ 4 ) 9 βˆ’ 2 𝑦 9 = 1 2 2
  • E ( π‘₯ βˆ’ 4 ) 3 βˆ’ 𝑦 4 = 1 2 2

Q3:

Suppose we model an asteroid’s trajectory by a hyperbolic path in the coordinate plane. The π‘₯ -axis is a line of symmetry of this hyperbola, and the object enters in the direction of 𝑦 = 1 3 π‘₯ βˆ’ 1 and leaves in the direction 𝑦 = βˆ’ 1 3 π‘₯ + 1 . The sun is positioned at the origin, and the object passes within 1 AU (astronomical unit) of the sun at its closest such that the sun is one focus of the hyperbola. Give the equation of the object’s path.

  • A ( π‘₯ βˆ’ 3 ) 4 βˆ’ 9 𝑦 4 = 1 2 2
  • B π‘₯ 9 βˆ’ ( 𝑦 βˆ’ 3 ) 4 = 1 2 2
  • C ( π‘₯ βˆ’ 3 ) 2 βˆ’ 3 𝑦 2 = 1 2 2
  • D ( π‘₯ βˆ’ 3 ) βˆ’ 𝑦 4 = 1 2 4 9 2
  • E ( π‘₯ βˆ’ 3 ) 9 βˆ’ 𝑦 4 = 1 2 2
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