text: From the point A (0, 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is :
  
  The given equation x² + 4x + ( y-3 )² = 0 can be written as
  
  ( x+2 )² + ( y-3 )² = 2².
  
  It represents the circle with C ≡ ( -2, 3 ) and r = 2.
  
  It touches Y-axis at the point A ≡ ( 0, 3 ).
  
  Let B( x₁, y₁ ) be a point on this circle so that
  
  (x₁ + 2)² + (y₁ - 3)² = 2², i.e.,
  
  (x₁)² + (y₁)² + 4(x₁) - 6(y₁) + 9 = 0 ......... (1)
  
  If M(h,k) is a point such that AM = 2(AB),
  
  i.e., AB = BM, then B is the mid-point of AM.
  
  Then, B ≡ ( (0+h)/2, (3+k)/2 ) ≡ ( x₁, y₁ )
  
  i.e., (h)/2 = x₁ ... and ... (3+k)/2 = y₁
  
  Hence, from (1),
  
  (h/2)² + [(3+k)/2]² + 4(h/2) - 6[(3+k)/2] + 9 = 0
  
  (h²/4) + [(3+k)²/4] + 2h - 3(3+k) + 9 = 0
  
  h² + (3+k)² + 8h - 12(3+k) + 36 = 0
  
  h² + 9 + 6k + k² + 8h - 36 - 12k + 36 = 0
  
  h² + k² + 8h - 6k + 9 = 0.
  
  Replacing the current co-ordinates (h,k) by the general co-ordinates (x,y),
  
  the required equation of the locus of M is
  
  x² + y² + 8x - 6y + 9 = 0
  
  that is ( x + 4 )² + ( y - 3 )² = 4².
explanation: The document contains a detailed mathematical problem and its solution. It starts with a question about finding the equation of the locus of point M, given certain conditions related to a circle and a chord. The document then proceeds to solve the problem step-by-step, leading to the final equation of the locus.
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: From the point A (0, 3) on the circle x² + 4x + (y - 3)² = 0 a chord AB is drawn & extended to a point M such that AM = 2 AB. The equation of the locus of M is :
  
  The given equation x² + 4x + ( y-3 )² = 0 can be written as
  
  ( x+2 )² + ( y-3 )² = 2².
  
  It represents the circle with C ≡ ( -2, 3 ) and r = 2.
  
  It touches Y-axis at the point A ≡ ( 0, 3 ).
  
  Let B( x₁, y₁ ) be a point on this circle so that
  
  (x₁ + 2)² + (y₁ - 3)² = 2², i.e.,
  
  (x₁)² + (y₁)² + 4(x₁) - 6(y₁) + 9 = 0 ......... (1)
  
  If M(h,k) is a point such that AM = 2(AB),
  
  i.e., AB = BM, then B is the mid-point of AM.
  
  Then, B ≡ ( (0+h)/2, (3+k)/2 ) ≡ ( x₁, y₁ )
  
  i.e., (h)/2 = x₁ ... and ... (3+k)/2 = y₁
  
  Hence, from (1),
  
  (h/2)² + [(3+k)/2]² + 4(h/2) - 6[(3+k)/2] + 9 = 0
  
  (h²/4) + [(3+k)²/4] + 2h - 3(3+k) + 9 = 0
  
  h² + (3+k)² + 8h - 12(3+k) + 36 = 0
  
  h² + 9 + 6k + k² + 8h - 36 - 12k + 36 = 0
  
  h² + k² + 8h - 6k + 9 = 0.
  
  Replacing the current co-ordinates (h,k) by the general co-ordinates (x,y),
  
  the required equation of the locus of M is
  
  x² + y² + 8x - 6y + 9 = 0
  
  that is ( x + 4 )² + ( y - 3 )² = 4².
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed mathematical problem and its solution. It starts with a question about finding the equation of the locus of point M, given certain conditions related to a circle and a chord. The document then proceeds to solve the problem step-by-step, leading to the final equation of the locus.
  
  Label: question-and-answer