Final answer:
The distance between the points is calculated using the distance formula, and the polar coordinates can be found using the formulas for r and theta.
Step-by-step explanation:
The distance between two points in a Cartesian plane can be found using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). Given the points A(1,-4) and B(3,4), we can substitute the coordinates into the formula and calculate the distance: d = sqrt((3 - 1)^2 + (4 - (-4))^2).
To find the polar coordinates of the points, we can use the formulas r = sqrt(x^2 + y^2) and θ = arctan(y/x). For point A(1,-4), r = sqrt(1^2 + (-4)^2) = sqrt(1 + 16) = sqrt(17) and θ = arctan((-4)/1) = arctan(-4) (in radians). Similarly, for point B(3,4), r = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 and θ = arctan(4/3).