Part 1 Given a directrix at x=-8 and a focus point at (-2,0), what are 5 points where the distance to the directrix is twice as far as the distance to the focus?
Example: (4,0) is one of the 5 points that I've found. What would some other examples be?
Then using algebra, find the equation of all points where the distance to the directrix is twice as far as the distance to the focus. After giving the equation, what is this locus called?
I'm highly confused as to how to find this equation and what the locus would be called.
Part 2 If the directrix were along the x-axis and the focus is at the point (2,-6), what would the equation of the locus be? How did you get this answer?
Again, not sure how to find an equation of a locus.
If $(x,y)$ is a point on that locus, then the condition you are given is that $x-(-8) = x+8$ (the distance to the directrix) is twice $\sqrt{(x+2)^2+y^2}$ (the distance to the focus). So if you set those two equal, square the resulting equation and simplify, you will get what looks like a reasonable equation for the locus. You can apply the same approach for your second question.