Question 1203355: Point a in the cartesian plane is at (2,3). Say that A is at the polar coordinates of (r, theta). If the cartesian coordinates of the point with polar coordinates (2r, theta + pi/2) are (x1, y1), and the cartesian coordinates of the point with polar coordinates (-r, -theta) are (x2,y2), what are x1,y1,x2, and y2?
Answer by greenestamps(13214)   (Show Source):
You can put this solution on YOUR website!
Given: A(2,3) = (r,theta)
(1) If B is (2r,theta+pi/2) then find B(x1,y1)
The angle theta+pi/2 is rotated pi/2 radians (90 degrees) counterclockwise from theta. Calling the origin O, the vector OA has slope 3/2, so vector OB (perpendicular to OA) has slope -2/3.
And B is twice as far from the origin as A.
Rotated 90 degrees from A(2,3) and twice as far from the origin puts B at (-6,4).
(I don't know why the "2r" displays as "2\r"....)
ANSWER: (x1,y1) = (-6,4)
(2) If C is (-r,-theta) then find C(x2,y2)
(r,-theta) is the reflection in the x-axis of A(2,3), so (r,-theta) is (2,-3).
(-r,-theta) is the reflection of (r,-theta) with respect to the origin, so (-r,-theta) is (-2,3).
ANSWER: (x2,y2) = (-2,3)
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