Worksheet: Equation of a Hyperbola

Practice

In this worksheet, we will practice analyzing and writing equations of hyperbolas.

Q1:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = 3 4 π‘₯ and 𝑦 = βˆ’ 3 4 π‘₯ , and its closest distance to the centre fountain is 20 yards. Find the equation of the hyperbola.

  • A π‘₯ 4 0 0 βˆ’ 𝑦 1 5 = 1 2 2
  • B π‘₯ 2 0 βˆ’ 𝑦 1 5 = 1 2 2
  • C π‘₯ 2 0 βˆ’ 𝑦 2 2 5 = 1 2 2
  • D π‘₯ 4 0 0 βˆ’ 𝑦 2 2 5 = 1 2 2
  • E π‘₯ 2 0 βˆ’ 𝑦 0 . 7 5 = 1 2 2

Q2:

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

Q3:

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

Q4:

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

Q5:

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 = π‘₯ βˆ’ 2 and leaves in the direction 𝑦 = βˆ’ π‘₯ + 2 . 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‘₯ + 2 ) βˆ’ 𝑦 = 1  
  • B π‘₯ βˆ’ ( 𝑦 βˆ’ 2 ) = 1  
  • C π‘₯ βˆ’ ( 𝑦 + 2 ) = 1  
  • D ( π‘₯ βˆ’ 2 ) βˆ’ 𝑦 = 1  
  • E ( π‘₯ βˆ’ 2 ) βˆ’ ( 𝑦 + 2 ) = 1  

Q6:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = 2 π‘₯ and 𝑦 = βˆ’ 2 π‘₯ , and its closest distance to the centre fountain is 6 yards. Find the equation of the hyperbola.

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

Q7:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a garden. The hedge will follow the asymptotes 𝑦 = π‘₯ and 𝑦 = βˆ’ π‘₯ , and its closest distance to the centre fountain is 5 yards. Find the equation of the hyperbola.

  • A π‘₯ 5 βˆ’ 𝑦 = 1 2 2
  • B π‘₯ 2 5 βˆ’ 𝑦 = 1 2 2
  • C π‘₯ 5 βˆ’ 𝑦 5 = 1 2 2
  • D π‘₯ 2 5 βˆ’ 𝑦 2 5 = 1 2 2
  • E π‘₯ βˆ’ 𝑦 2 5 = 1 2 2

Q8:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the centre of a yard. The hedge will follow the asymptotes 𝑦 = 1 2 π‘₯ and 𝑦 = βˆ’ 1 2 π‘₯ , and its closest distance to the centre fountain is 10 yards. Find the equation of the hyperbola.

  • A π‘₯ 1 0 0 βˆ’ 𝑦 0 . 2 5 = 1 2 2
  • B π‘₯ 1 0 βˆ’ 𝑦 0 . 5 = 1 2 2
  • C π‘₯ 1 0 βˆ’ 𝑦 5 = 1 2 2
  • D π‘₯ 1 0 0 βˆ’ 𝑦 2 5 = 1 2 2
  • E π‘₯ 2 0 βˆ’ 𝑦 1 0 = 1 2 2

Q9:

Write the equation of the rectangular hyperbola passing through ( 1 , 1 ) with asymptotes meeting at ( 3 , βˆ’ 4 ) .

  • A 𝑦 = βˆ’ 1 0 π‘₯ + 4 + 3
  • B 𝑦 = βˆ’ 2 4 π‘₯ βˆ’ 4 βˆ’ 7
  • C 𝑦 = 2 0 π‘₯ + 3 βˆ’ 4
  • D 𝑦 = βˆ’ 1 0 π‘₯ βˆ’ 3 βˆ’ 4
  • E 𝑦 = 1 π‘₯ βˆ’ 5 βˆ’ 9

Q10:

Suppose that we model an object’s trajectory in the solar system by a hyperbolic path in the coordinate plane, with its origin at the sun and its units in astronomical units (AU). The π‘₯ -axis is a line of symmetry of this hyperbola. The object enters in the direction of 𝑦 = 3 π‘₯ βˆ’ 9 , leaves in the direction of 𝑦 = βˆ’ 3 π‘₯ + 9 , and passes within 1 AU of the sun at its closest point. Using the equations of the asymptotes, find the equation of the object’s path.

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

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