How to figure out what $|z-1|+|z+5| < 10$ represents without calculations? 
How to figure out what $|z-1|+|z+5| < 10$ represents without calculations?

Draw  a set of given complex numbers $|z-1|+|z+5| < 10$.
I know what $|z-1|< 10$ means separately, and intuitively I would draw two circles with their centers respectively. Now I get lost, because suddenly I have no idea what the solution would be.
So I looked up the solution and that's actually an ellipse. Any good explanation?
 A: Recall that the definition of an ellipse is that for a point on the ellipse, the sum of the distances from the foci is constant.
In this case, your foci are $1$ and $-5$ and $|z-1|+|z+5|=10$ is the ellipse where the sum is $10$.  
Now, from the inequality, you want the interior of the ellipse.
A: Hint: An ellipse is the locus of all the points in a plane the sum of whose distances from two fixed points (also called the foci) is a constant. See this for more details.
In your problem, the foci are $(1,0)$ and $(-5,0)$.
A: $|z-1|$ is the distance from $z$ to $1+0i$.  You are given that the sum of the distances from two points to $z$ is less than $10$.  This is the definition of the interior of an ellipse.  The foci are $1+0i$ and $-5+0i$, the semimajor axis is $3$ and is along the real axis.
A: You might not remember this from high school, but you can think of an ellipse as the set of points in the plane whose sum of distances to two distinguished foci is constant. That's all that's going on here, where the foci are at $1$ and $-5$, and that constant is 10. Of course, the inequality just means we take all the points in the plane inside the ellipse, excluding the boundary. 
A: Sum of distances from z to two points is less than 10.The boudry is an ellipce with the points 1 and -5 at its foci.Find the center and the rest is easy.Remember that only the interior of the ellipse is described by your equation.
