Final answer:
Without a given ratio, the coordinates of point P cannot be found. To determine distance and polar coordinates for points A and B, we use the Pythagorean theorem and trigonometric functions, respectively.
Step-by-step explanation:
Finding the Point P On a Directed Line Segment
To find the point P that divides the directed line segment from point A to point B in a given ratio, we can use the section formula. However, since no particular ratio is given in this question, we cannot provide the coordinates of point P without additional information. Typically, the formula involves taking a weighted average of the x-coordinates and y-coordinates of points A and B using the given ratio.
Distance Between Points A and B
The formula to find the distance between two points A(x1, y1) and B(x2, y2) in the Cartesian coordinate system is derived from the Pythagorean theorem. The distance d is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Finding Polar Coordinates
To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), we use the equations:
r = √(x^2 + y^2)
θ = atan2(y, x), where atan2 is the arctangent function that considers the signs of x and y.
For the points given in the question, additional steps such as finding angles or using trigonometric relationships would be necessary to provide the polar coordinates.