Graph Curves in the Complex Plane

[msd213]

The problem statement, all variables and given/known data[/b]

Graph the locus represented by the following.

\left|z+2i\right| + \left|z-2i\right| = 6

2. Relevant equations



3. The attempt at a solution

z = x + iy so

z-2i = x + (y-2)i and z+2i = x + (y-2)i

So I have:

sqrt(x^2 + (y-2)^2) + sqrt(x^2 + (y+2)^2) = 6

This seems correct but I don't know how to put this in a more manageable form to graph it.

[Mark44]

The problem statement, all variables and given/known data[/b]

Graph the locus represented by the following.

\left|z+2i\right| + \left|z-2i\right| = 6

2. Relevant equations



3. The attempt at a solution

z = x + iy so

z-2i = x + (y-2)i and z+2i = x + (y-2)i

So I have:

sqrt(x^2 + (y-2)^2) + sqrt(x^2 + (y+2)^2) = 6

This seems correct but I don't know how to put this in a more manageable form to graph it.

Move one radical to the other side, then square both sides. After doing this, you'll be able to eliminate several terms, and do some other simplification to get one radical on one side. Square both sides again to get this in final form.

The graph you get should be an ellipse with 2i and -2i as the foci.

[HallsofIvy]

Move one radical to the other side, then square both sides. After doing this, you'll be able to eliminate several terms, and do some other simplification to get one radical on one side. Square both sides again to get this in final form.

The graph you get should be an ellipse with 2i and -2i as the foci.

Mark44 knows this because he knows that |a- b|, geometrically, is the distance between the points a and b in the complex plane, and that an ellipse is defined by the property that the total distance from a point on the ellipse to the two foci is a constant.

[msd213]

Mark44 knows this because he knows that |a- b|, geometrically, is the distance between the points a and b in the complex plane, and that an ellipse is defined by the property that the total distance from a point on the ellipse to the two foci is a constant.

Oh! I see now, thank you.

[Mark44]

Mark44 knows this because he knows that |a- b|, geometrically, is the distance between the points a and b in the complex plane, and that an ellipse is defined by the property that the total distance from a point on the ellipse to the two foci is a constant.
Right. One definition of an ellipse is that it is the locus of points P such that the sum of the distances from P to two fixed points (the foci) is constant.