sqrt[(x-2)^2 + (y+1)^2] + sqrt[(x-10)^2 + (y-5)^2] = 10

Now the two radicands describe the distance from (2,-1) to (x,y) and (10,5) to (x,y).  So what we actually have is an ellipse with foci of (2,-1) and (10,5) and the sum of distances equals 10.

But the distance from (2,-1) to (10,5) is also 10.  This means our ellipse is a degenerate ellipse consisting solely of the line segment from (2,-1) to (10,5).

The equation of the line passing through (2,-1) and (10,5) is 3x-4y=10.  Then the intersection of this line with 3x+4y=26 is (6,2).  This does lay on the segment spanning (2,-1) to (10,5), so it is a valid solution.