text: #### Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity   Option 1) 7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0 Option 2) 7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0 Option 3) 7x2 - 2y2 + xy - 14x + 2y - 22 = 0 Option 4) none of the above
  
  Eccentricity -
  
  The ratio of a distance of point from focus to distance from fixed line.
  
  - wherein
  
  It is donated by .
  
  &
  
  Directrix -
  
  The fixed straight line of a conic section.
  
  - wherein
  
  &
  
  Focus -
  
  The fixed point of a conic section
  
  - wherein
  
  Option 1)
  
  7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0
  
  This solution is correct
  
  Option 2)
  
  7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0
  
  This solution is incorrect
  
  Option 3)
  
  7x2 - 2y2 + xy - 14x + 2y - 22 = 0
  
  This solution is incorrect
  
  Option 4)
  
  none of the above
  
  This solution is incorrect
explanation: The document contains a math question asking for the equation of a hyperbola given its directrix, focus, and eccentricity. It also provides multiple-choice options and indicates which option is correct.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: #### Find the equation of the hyperbola whose directrix is 2x + y = 1, focus (1, 2) and eccentricity   Option 1) 7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0 Option 2) 7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0 Option 3) 7x2 - 2y2 + xy - 14x + 2y - 22 = 0 Option 4) none of the above
  
  Eccentricity -
  
  The ratio of a distance of point from focus to distance from fixed line.
  
  - wherein
  
  It is donated by .
  
  &
  
  Directrix -
  
  The fixed straight line of a conic section.
  
  - wherein
  
  &
  
  Focus -
  
  The fixed point of a conic section
  
  - wherein
  
  Option 1)
  
  7x2 - 2y2 + 12xy - 2x + 14y - 22 = 0
  
  This solution is correct
  
  Option 2)
  
  7x2 - 2y2 + 2xy - 2x + 14y - 22 = 0
  
  This solution is incorrect
  
  Option 3)
  
  7x2 - 2y2 + xy - 14x + 2y - 22 = 0
  
  This solution is incorrect
  
  Option 4)
  
  none of the above
  
  This solution is incorrect
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question asking for the equation of a hyperbola given its directrix, focus, and eccentricity. It also provides multiple-choice options and indicates which option is correct.
  
  Label: question-and-answer