Question 202346
in general,  this  type  of  problem  require  that  we  complete  the  square.
Remember,,separate  the  x^2 and x^1 terms,,,x^2 must  have  coefficient =1,,take  1/2  od  x^1 coefficient,,,and  square  it  for  third  term.,,,then  add  compensating term  to  equation  to  balance.
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x^2 -4x   -4y^2 -24y =48
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{x^2 -4x }-4{y^2 +6y  }  =48
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{x^2 -4x -4} -4 {y^2 +6y +9} = 48 -4 -36 = 8,,,,,,(-4*9 = -36)
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{ x-2}^2 -4 {y+3}^2 =8
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divide  thru  by  8
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{(x-2)^2}/8 -{ (y+3)^2}/2 =1
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This  is  the  std  form  of  a  hyperbola
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(x-h)^2 /a^2 - (y-k)^2/b^2 =1