Worksheet: Equation of a Hyperbola

In this worksheet, we will practice writing the equation of a hyperbola and finding a hyperbola's vertices and foci using its equation and graph.

Q1:

A gardener wants to grow a hedge around a fountain in their yard. They decide to plant the hedge along a hyperbola, where one of the foci of the hyperbola is at the fountain. At its closest, the hedge will be a distance of 20 yards from the fountain. Using a coordinate system whose origin is at the fountain and with units in yards, the path of the hedge has asymptotes 𝑦=34π‘₯ and 𝑦=βˆ’34π‘₯. Find the equation of the hyperbola in this coordinate system.

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

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=23π‘₯ and 𝑦=βˆ’23π‘₯, and its closest distance to the center fountain is 12 yards. Find the equation of the hyperbola.

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

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=0.5π‘₯+2 and leaves in the direction 𝑦=βˆ’0.5π‘₯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 ) 9 βˆ’ 2 𝑦 9 = 1  
  • B ( π‘₯ βˆ’ 4 ) 3 βˆ’ 𝑦 4 = 1  
  • C ( π‘₯ + 4 ) 9 βˆ’ 4 𝑦 9 = 1  
  • D ( π‘₯ + 4 ) βˆ’ 𝑦 1 6 = 1   οŠͺ 
  • E ( π‘₯ + 4 ) 3 βˆ’ 𝑦 7 = 1  

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=13π‘₯βˆ’1 and leaves in the direction 𝑦=βˆ’13π‘₯+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 ) βˆ’ 𝑦 4 = 1  οŠͺ  
  • B ( π‘₯ βˆ’ 3 ) 4 βˆ’ 9 𝑦 4 = 1  
  • C π‘₯ 9 βˆ’ ( 𝑦 βˆ’ 3 ) 4 = 1  
  • D ( π‘₯ βˆ’ 3 ) 2 βˆ’ 3 𝑦 2 = 1  
  • E ( π‘₯ βˆ’ 3 ) 9 βˆ’ 𝑦 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‘₯ βˆ’ 2 ) βˆ’ 𝑦 = 1  
  • B ( π‘₯ βˆ’ 2 ) βˆ’ ( 𝑦 + 2 ) = 1  
  • C π‘₯ βˆ’ ( 𝑦 βˆ’ 2 ) = 1  
  • D ( π‘₯ + 2 ) βˆ’ 𝑦 = 1  
  • E π‘₯ βˆ’ ( 𝑦 + 2 ) = 1  

Q6:

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

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

Q7:

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

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

Q8:

A hedge is to be constructed in the shape of a hyperbola near a fountain at the center of a yard. The hedge will follow the asymptotes 𝑦=12π‘₯ and 𝑦=βˆ’12π‘₯, and its closest distance to the center fountain is 10 yards. Find the equation of the hyperbola.

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

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

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=3π‘₯βˆ’9, leaves in the direction of 𝑦=βˆ’3π‘₯+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 ) 4 βˆ’ 𝑦 3 6 = 1  
  • B ( π‘₯ βˆ’ 3 ) 2 βˆ’ 𝑦 6 = 1  
  • C ( π‘₯ βˆ’ 3 ) 6 βˆ’ 𝑦 2 = 1  
  • D ( π‘₯ βˆ’ 3 ) 9 βˆ’ 𝑦 4 = 1  
  • E π‘₯ 3 6 βˆ’ ( 𝑦 βˆ’ 3 ) 4 = 1  

Q11:

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=2π‘₯βˆ’2 and leaves in the direction 𝑦=βˆ’2π‘₯+2. The sun is positioned at the origin, and the object passes within 0.5 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’ 1 ) 0 . 5 βˆ’ 𝑦 = 1  
  • B π‘₯ 0 . 5 βˆ’ ( 𝑦 βˆ’ 1 ) = 1  
  • C π‘₯ 4 βˆ’ ( 𝑦 βˆ’ 1 ) 0 . 7 5 = 1  
  • D ( π‘₯ βˆ’ 1 ) 0 . 2 5 βˆ’ 𝑦 0 . 7 5 = 1  
  • E ( π‘₯ βˆ’ 1 ) 0 . 2 5 βˆ’ 𝑦 = 1  

Q12:

The graph shows a sketch of the hyperbola given by the equation 4π‘¦βˆ’π‘₯+8π‘¦βˆ’10π‘₯=25.

Give the coordinates of the center 𝐢.

  • A 𝐢 ( βˆ’ 5 , βˆ’ 1 )
  • B 𝐢 ( βˆ’ 1 , βˆ’ 5 )
  • C 𝐢 ( 1 , 5 )
  • D 𝐢 ( 5 , 1 )

Give the coordinates of the vertices π‘‰οŠ§ and π‘‰οŠ¨.

  • A 𝑉 ( βˆ’ 3 , βˆ’ 1 )  , 𝑉 ( βˆ’ 7 , βˆ’ 1 ) 
  • B 𝑉 ( βˆ’ 5 , 0 )  , 𝑉 ( βˆ’ 5 , βˆ’ 2 ) 
  • C 𝑉 ( βˆ’ 4 , βˆ’ 1 )  , 𝑉 ( βˆ’ 6 , βˆ’ 1 ) 
  • D 𝑉 ( βˆ’ 5 , 1 )  , 𝑉 ( βˆ’ 5 , βˆ’ 3 ) 

Give the coordinates of the foci 𝐹 and 𝐹.

  • A 𝐹 ο€» βˆ’ 5 + √ 5 , βˆ’ 1   , 𝐹 ο€» βˆ’ 5 βˆ’ √ 5 , βˆ’ 1  
  • B 𝐹 ο€» βˆ’ 5 , βˆ’ 1 + √ 5   , 𝐹 ο€» βˆ’ 5 , βˆ’ 1 βˆ’ √ 5  
  • C 𝐹 ο€» 5 , βˆ’ 1 + √ 5   , 𝐹 ο€» 5 , βˆ’ 1 βˆ’ √ 5  
  • D 𝐹 ο€» 5 + √ 5 , βˆ’ 1   , 𝐹 ο€» 5 βˆ’ √ 5 , βˆ’ 1  

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A 𝐴 𝑦 + 1 = 1 2 ( π‘₯ + 5 )  : , 𝐴 𝑦 + 1 = βˆ’ 1 2 ( π‘₯ + 5 )  :
  • B 𝐴 𝑦 + 1 = 2 ( π‘₯ + 5 )  : , 𝐴 𝑦 + 1 = βˆ’ 2 ( π‘₯ + 5 )  :
  • C 𝐴 𝑦 + 1 = 1 2 ( π‘₯ βˆ’ 5 )  : , 𝐴 𝑦 + 1 = βˆ’ 1 2 ( π‘₯ βˆ’ 5 )  :
  • D 𝐴 𝑦 βˆ’ 1 = 1 2 ( π‘₯ + 5 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 1 2 ( π‘₯ + 5 )  :

Q13:

The graph shows a sketch of the hyperbola given by the equation (π‘¦βˆ’2)25βˆ’(π‘₯+3)2=1.

Give the coordinates of the center 𝐢.

  • A 𝐢 ( 2 , βˆ’ 3 )
  • B 𝐢 ( βˆ’ 3 , 2 )
  • C 𝐢 ( βˆ’ 2 , 3 )
  • D 𝐢 ( 3 , βˆ’ 2 )

Give the coordinates of the vertices π‘‰οŠ§ and π‘‰οŠ¨.

  • A 𝑉 ( 2 , 2 )  , 𝑉 ( βˆ’ 8 , 2 ) 
  • B 𝑉 ο€» βˆ’ 3 , 2 + √ 2   , 𝑉 ο€» βˆ’ 3 , 2 βˆ’ √ 2  
  • C 𝑉 ο€» βˆ’ 3 + √ 2 , 2   , 𝑉 ο€» βˆ’ 3 βˆ’ √ 2 , 2  
  • D 𝑉 ( βˆ’ 3 , 7 )  , 𝑉 ( βˆ’ 3 , βˆ’ 3 ) 

Give the coordinates of the foci 𝐹 and 𝐹.

  • A 𝐹 ο€» βˆ’ 3 , 2 + 3 √ 3   , 𝐹 ο€» βˆ’ 3 , 2 βˆ’ 3 √ 3  
  • B 𝐹 ο€» 3 + 3 √ 3 , 2   , 𝐹 ο€» 3 βˆ’ 3 √ 3 , 2  
  • C 𝐹 ο€» 3 , 2 + 3 √ 3   , 𝐹 ο€» 3 , 2 βˆ’ 3 √ 3  
  • D 𝐹 ο€» βˆ’ 3 + 3 √ 3 , 2   , 𝐹 ο€» βˆ’ 3 βˆ’ 3 √ 3 , 2  

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A 𝐴 𝑦 βˆ’ 2 = 5 √ 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 βˆ’ 2 = βˆ’ 5 √ 2 ( π‘₯ βˆ’ 3 )  :
  • B 𝐴 𝑦 βˆ’ 2 = 5 √ 2 ( π‘₯ + 3 )  : , 𝐴 𝑦 βˆ’ 2 = βˆ’ 5 √ 2 ( π‘₯ + 3 )  :
  • C 𝐴 𝑦 + 2 = 5 √ 2 ( π‘₯ + 3 )  : , 𝐴 𝑦 + 2 = βˆ’ 5 √ 2 ( π‘₯ + 3 )  :
  • D 𝐴 𝑦 βˆ’ 2 = √ 2 5 ( π‘₯ + 3 )  : , 𝐴 𝑦 βˆ’ 2 = βˆ’ √ 2 5 ( π‘₯ + 3 )  :

Q14:

The graph shows a sketch of the hyperbola given by the equation (π‘₯βˆ’3)4βˆ’(π‘¦βˆ’1)16=1.

Give the coordinates of the center 𝐢.

  • A 𝐢 ( 3 , 1 )
  • B 𝐢 ( 1 , 3 )
  • C 𝐢 ( βˆ’ 1 , βˆ’ 3 )
  • D 𝐢 ( βˆ’ 3 , βˆ’ 1 )

Give the coordinates of the vertices π‘‰οŠ§ and π‘‰οŠ¨.

  • A 𝑉 ( 5 , 3 )  , 𝑉 ( βˆ’ 3 , 3 ) 
  • B 𝑉 ( 5 , 1 )  , 𝑉 ( 1 , 1 ) 
  • C 𝑉 ( 3 , 3 )  , 𝑉 ( βˆ’ 1 , 3 ) 
  • D 𝑉 ( 7 , 1 )  , 𝑉 ( βˆ’ 1 , 1 ) 

Give the coordinates of the foci 𝐹 and 𝐹.

  • A 𝐹 ο€» βˆ’ 1 + 2 √ 5 , 3   , 𝐹 ο€» βˆ’ 1 βˆ’ 2 √ 5 , 3  
  • B 𝐹 ο€» βˆ’ 3 + 2 √ 5 , 1   , 𝐹 ο€» βˆ’ 3 βˆ’ 2 √ 5 , 1  
  • C 𝐹 ο€» 3 + 2 √ 5 , 1   , 𝐹 ο€» 3 βˆ’ 2 √ 5 , 1  
  • D 𝐹 ο€» 1 + 2 √ 5 , 3   , 𝐹 ο€» 1 βˆ’ 2 √ 5 , 3  

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A 𝐴 𝑦 βˆ’ 1 = 2 ( π‘₯ + 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 2 ( π‘₯ + 3 )  :
  • B 𝐴 𝑦 βˆ’ 1 = 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 2 ( π‘₯ βˆ’ 3 )  :
  • C 𝐴 𝑦 βˆ’ 1 = 1 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 βˆ’ 1 = βˆ’ 1 2 ( π‘₯ βˆ’ 3 )  :
  • D 𝐴 𝑦 + 1 = 2 ( π‘₯ βˆ’ 3 )  : , 𝐴 𝑦 + 1 = βˆ’ 2 ( π‘₯ βˆ’ 3 )  :

Q15:

The graph shows a sketch of the hyperbola given by the equation 4π‘₯βˆ’9π‘¦βˆ’16π‘₯βˆ’182𝑦=29.

Give the coordinates of the center 𝐢.

  • A 𝐢 ( βˆ’ 2 , 1 )
  • B 𝐢 ( βˆ’ 1 , 2 )
  • C 𝐢 ( 2 , βˆ’ 1 )
  • D 𝐢 ( 1 , βˆ’ 2 )

Give the coordinates of the vertices π‘‰οŠ§ and π‘‰οŠ¨.

  • A 𝑉 ( 5 , βˆ’ 1 ) , 𝑉 ( βˆ’ 1 , βˆ’ 1 )  
  • B 𝑉 ( 4 , βˆ’ 1 ) , 𝑉 ( 0 , βˆ’ 1 )  
  • C 𝑉 ( 2 , 2 ) , 𝑉 ( βˆ’ 4 , 2 )  
  • D 𝑉 ( 1 , 2 ) , 𝑉 ( βˆ’ 3 , 2 )  

Give the coordinates of the foci 𝐹 and 𝐹.

  • A 𝐹 ο€» 2 + √ 1 3 , βˆ’ 1  , 𝐹 ο€» 2 βˆ’ √ 1 3 , βˆ’ 1   
  • B 𝐹 ο€» βˆ’ 2 , βˆ’ 1 + √ 1 3  , 𝐹 ο€» βˆ’ 2 , βˆ’ 1 βˆ’ √ 1 3   
  • C 𝐹 ο€» βˆ’ 2 + √ 1 3 , βˆ’ 1  , 𝐹 ο€» βˆ’ 2 βˆ’ √ 1 3 , βˆ’ 1   
  • D 𝐹 ο€» 2 , βˆ’ 1 + √ 1 3  , 𝐹 ο€» 2 , βˆ’ 1 βˆ’ √ 1 3   

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A 𝐴 ∢ 𝑦 + 1 = 3 2 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 3 2 ( π‘₯ βˆ’ 2 )  
  • B 𝐴 ∢ 𝑦 βˆ’ 1 = 2 3 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 βˆ’ 1 = βˆ’ 2 3 ( π‘₯ βˆ’ 2 )  
  • C 𝐴 ∢ 𝑦 + 1 = 2 3 ( π‘₯ βˆ’ 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 2 3 ( π‘₯ βˆ’ 2 )  
  • D 𝐴 ∢ 𝑦 + 1 = 2 3 ( π‘₯ + 2 ) , 𝐴 ∢ 𝑦 + 1 = βˆ’ 2 3 ( π‘₯ + 2 )  

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.