Question

If the polar coordinates of a point are (r,φ) and its rectangular coordinates are (x,y), determine the polar coordinates of the following points: (a) (−x, y), (b) (−2x, −2y), and (c) (3x, −3y).

a) (r,ϕ),(−r,ϕ+π)
b) (r,ϕ),(2r,ϕ+π)
c) (r,ϕ),(3r,ϕ+π)
d) (r,ϕ),(−3r,ϕ+π)

Answer 1

The polar coordinates for the given points (a) (-x, y), (b) (-2x, -2y), and (c) (3x, -3y) are respectively (a) (r, φ + π), (b) (2r, φ + π), and (c) (3r, φ + π), obtained by adjusting the radius and adding π to the angle when the signs of the rectangular coordinates change.

Step-by-step explanation:

To determine the polar coordinates of the given points, one must understand the relationship between rectangular and polar coordinates. If (x, y) are the rectangular coordinates of a point, and (r, φ) are the polar coordinates, then:

• x = r × cos(φ)
• y = r × sin(φ)

For the points in question:

1. (−x, y): Changing the sign of x in the rectangular system corresponds to adding π to the angle φ in polar coordinates if r is positive. The polar coordinates would be (r, φ + π).
2. (−2x, −2y): Multiplying both x and y by -2 scales the original radial distance r by 2 and changes the angle by π. Hence, the polar coordinates are (2r, φ + π).
3. (3x, −3y): Multiplying x by 3 and y by -3 scales the radial distance r by 3 while changing the sign of y indicates adding π to the angle φ. Thus, the polar coordinates are (3r, φ + π).

The correct options are:

• (a) (r, φ + π)
• (b) (2r, φ + π)
• (c) (3r, φ + π)