Lesson Worksheet: Equation of a Hyperbola Mathematics

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:

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

  • A𝑦=−10𝑥−3−4
  • B𝑦=20𝑥+3−4
  • C𝑦=−10𝑥+4+3
  • D𝑦=1𝑥−5−9
  • E𝑦=−24𝑥−4−7

Q2:

The graph shows a sketch of the hyperbola given by the equation 4𝑦−𝑥+8𝑦−10𝑥=25.

Give the coordinates of the center 𝐶.

  • A𝐶(−5,−1)
  • B𝐶(−1,−5)
  • C𝐶(1,5)
  • D𝐶(5,1)

Give the coordinates of the vertices 𝑉 and 𝑉.

  • A𝑉(−3,−1), 𝑉(−7,−1)
  • B𝑉(−5,0), 𝑉(−5,−2)
  • C𝑉(−4,−1), 𝑉(−6,−1)
  • D𝑉(−5,1), 𝑉(−5,−3)

Give the coordinates of the foci 𝐹 and 𝐹.

  • A𝐹−5+√5,−1, 𝐹−5−√5,−1
  • B𝐹−5,−1+√5, 𝐹−5,−1−√5
  • C𝐹5,−1+√5, 𝐹5,−1−√5
  • D𝐹5+√5,−1, 𝐹5−√5,−1

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A𝐴𝑦+1=12(𝑥+5):, 𝐴𝑦+1=−12(𝑥+5):
  • B𝐴𝑦+1=2(𝑥+5):, 𝐴𝑦+1=−2(𝑥+5):
  • C𝐴𝑦+1=12(𝑥−5):, 𝐴𝑦+1=−12(𝑥−5):
  • D𝐴𝑦−1=12(𝑥+5):, 𝐴𝑦−1=−12(𝑥+5):

Q3:

The graph shows a sketch of the hyperbola given by the equation (𝑦−2)25−(𝑥+3)2=1.

Give the coordinates of the center 𝐶.

  • A𝐶(2,−3)
  • B𝐶(−3,2)
  • C𝐶(−2,3)
  • D𝐶(3,−2)

Give the coordinates of the vertices 𝑉 and 𝑉.

  • A𝑉(2,2), 𝑉(−8,2)
  • B𝑉−3,2+√2, 𝑉−3,2−√2
  • C𝑉−3+√2,2, 𝑉−3−√2,2
  • D𝑉(−3,7), 𝑉(−3,−3)

Give the coordinates of the foci 𝐹 and 𝐹.

  • A𝐹−3,2+3√3, 𝐹−3,2−3√3
  • B𝐹3+3√3,2, 𝐹3−3√3,2
  • C𝐹3,2+3√3, 𝐹3,2−3√3
  • D𝐹−3+3√3,2, 𝐹−3−3√3,2

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A𝐴𝑦−2=5√2(𝑥−3):, 𝐴𝑦−2=−5√2(𝑥−3):
  • B𝐴𝑦−2=5√2(𝑥+3):, 𝐴𝑦−2=−5√2(𝑥+3):
  • C𝐴𝑦+2=5√2(𝑥+3):, 𝐴𝑦+2=−5√2(𝑥+3):
  • D𝐴𝑦−2=√25(𝑥+3):, 𝐴𝑦−2=−√25(𝑥+3):

Q4:

The graph shows a sketch of the hyperbola given by the equation (𝑥−3)4−(𝑦−1)16=1.

Give the coordinates of the center 𝐶.

  • A𝐶(3,1)
  • B𝐶(1,3)
  • C𝐶(−1,−3)
  • D𝐶(−3,−1)

Give the coordinates of the vertices 𝑉 and 𝑉.

  • A𝑉(5,3), 𝑉(−3,3)
  • B𝑉(5,1), 𝑉(1,1)
  • C𝑉(3,3), 𝑉(−1,3)
  • D𝑉(7,1), 𝑉(−1,1)

Give the coordinates of the foci 𝐹 and 𝐹.

  • A𝐹−1+2√5,3, 𝐹−1−2√5,3
  • B𝐹−3+2√5,1, 𝐹−3−2√5,1
  • C𝐹3+2√5,1, 𝐹3−2√5,1
  • D𝐹1+2√5,3, 𝐹1−2√5,3

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A𝐴𝑦−1=2(𝑥+3):, 𝐴𝑦−1=−2(𝑥+3):
  • B𝐴𝑦−1=2(𝑥−3):, 𝐴𝑦−1=−2(𝑥−3):
  • C𝐴𝑦−1=12(𝑥−3):, 𝐴𝑦−1=−12(𝑥−3):
  • D𝐴𝑦+1=2(𝑥−3):, 𝐴𝑦+1=−2(𝑥−3):

Q5:

The graph shows a sketch of the hyperbola given by the equation 4𝑥−9𝑦−16𝑥−182𝑦=29.

Give the coordinates of the center 𝐶.

  • A𝐶(−2,1)
  • B𝐶(−1,2)
  • C𝐶(2,−1)
  • D𝐶(1,−2)

Give the coordinates of the vertices 𝑉 and 𝑉.

  • A𝑉(5,−1),𝑉(−1,−1)
  • B𝑉(4,−1),𝑉(0,−1)
  • C𝑉(2,2),𝑉(−4,2)
  • D𝑉(1,2),𝑉(−3,2)

Give the coordinates of the foci 𝐹 and 𝐹.

  • A𝐹2+√13,−1,𝐹2−√13,−1
  • B𝐹−2,−1+√13,𝐹−2,−1−√13
  • C𝐹−2+√13,−1,𝐹−2−√13,−1
  • D𝐹2,−1+√13,𝐹2,−1−√13

Give the equations of the asymptotes 𝐴 and 𝐴.

  • A𝐴∶𝑦+1=32(𝑥−2),𝐴∶𝑦+1=−32(𝑥−2)
  • B𝐴∶𝑦−1=23(𝑥−2),𝐴∶𝑦−1=−23(𝑥−2)
  • C𝐴∶𝑦+1=23(𝑥−2),𝐴∶𝑦+1=−23(𝑥−2)
  • D𝐴∶𝑦+1=23(𝑥+2),𝐴∶𝑦+1=−23(𝑥+2)

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