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Problem 6 a: Solve the equations x^2/9+y^2 /4=1, x^2-y^2=1 for {x,y}. b: Use NSolve to find approximate solutions to this set of equations. c: Now use 2-dimensional zooming and ContourPlot to find an approximation rounded to two places after the decimal point to this pair of equations.

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Clear["Global`*"]

eqns = {x^2/9 + y^2/4 == 1, x^2 - y^2 == 1};

sol = NSolve[eqns, {x, y}]

(* {{x -> -1.86052, y -> -1.56893}, {x -> 1.86052, y -> 1.56893}, {x -> 1.86052, 
  y -> -1.56893}, {x -> -1.86052, y -> 1.56893}} *)

Verifying the solutions,

And @@@ (eqns /. sol)

(* {True, True, True, True} *)

pts = {x, y} /. sol;

Manipulate[
 ContourPlot[
  Evaluate@eqns,
  {x, pts[[loc, 1]] - d, pts[[loc, 1]] + d},
  {y, pts[[loc, 2]] - d, pts[[loc, 2]] + d},
  Epilog -> {Red, AbsolutePointSize[6],
    Point[{x, y} /. sol]},
  ImageSize -> Medium],
 {{loc, 1, "Intersection"}, Range[4]},
 {{d, 5, "Zoom"}, 0.01, 5, 0.01, Appearance -> "Labeled"}]

enter image description here

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