The equation is
i) Let and be fixed. Describe the set of points satisfying the above equation for every possible choice of and .
ii) Now let and, using a rotation of the plane, describe the locus of points satisfying the above equation.
I tried algebra for both i) and ii) and just got a big mess, no matter what form I converted the complex number to, or even just matching up real and imaginary parts.
I am not sure how else to find the set of that would satisfy the above equation.
I am also lost by what they mean by locus in ii), I am assuming it must be some sort of conic section, either a hyperbola or an ellipse. But I dont know how to show this either.
Any advice, nudge or hint in the right direction would be helpful. Thanks for answering!
So it's a hyperbola when ?
I guess it would be a parabola when ? And if it's a ellipse?
And I must find the equation of this hyperbola? something to do with and ?
is the length from the foci to the center and is the length from the vertex to the center. What would this be centered at though? The origin?
Darn, I wish I remembered the formula for a hyperbola. I am sure and are used somehow.
But, c is greater than zero. I assuming it'd be the right branch. I am not sure how my equations are wrong. Is there another approach that is not algebraic? Could you explain it please? I've tried to draw pictures of these things, but nothing makes sense.
Ray from z = -a in direction of negative real axis if c = a.
No solution if c > a.
This all follows from the geometric approach suggested in post #2.
I suggest you consider some concrete examples:
a > c: |z - 2| - |z + 2| = 1
a = c: |z - 2| - |z + 2| = 4
a < c: |z - 2| - |z + 2| = 6
If you're uncomfortable with the geometric approach, substitute z = x + iy and take an algebraic approach.
Since you have not shown any working as to how you got your answers, it's impossible to know the mistakes you have made in getting the wrong equations. You will need to show detaield working if your work is to be reviewed properly.
Note: I corrected a typo in my first post.
There is an implicit restriction in the line I labelled **, namely . This restriction needs to be imposed once you have cartesian equation for the hyperbola. That's why only one branch is defined.