### Instruction: The graph of an equation \[\sqrt{(x-3)^2 + (y+4)^2} + \sqrt{(x+5)^2 + (y-8)^2} = 20.\]is an ellipse. What is the distance between its foci?### Response: The equation looks complicated, but I notice that it has two square roots, each containing a term of the form \[(x-h)^2 + (y-k)^2\]which is the distance from the point (x, y) to the point (h, k).
So I can rewrite the equation as \[d_1 + d_2 = 20,\]where \(d_1\) is the distance from (x, y) to (3, -4) and \(d_2\) is the distance from (x, y) to (-5, 8).
This means that the points (3, -4) and (-5, 8) are the foci of the ellipse, and the sum of the distances from any point on the ellipse to the foci is always 20.