SetAttributes[c,  Constant]

Eliminate[{x^2 - c y^2 == 1,  Dt[x^2 - c y^2, x] == 0}, 
      c]

Eliminate[{y == c/x^2 + c,  Dt[y - c/x^2, x] == 0}, 
      c]

<< Graphics`ImplicitPlot`

ImplicitPlot[
      Evaluate[Table[
          x^2 + (y - c)^2 == 3^2,  {c,  -10, 10, 
            1}]],  {x,  -5,  5},  
      PlotStyle -> 
        Table[Hue[c/20],  {c,  -10,  10,  1}]]

Eliminate[{x^2 + (y - c)^2 == 3^2,  
        Dt[x^2 + (y - c)^2, x] == 0}, c]