Question

A curve, described by x2 + y2 + 12y = 0, has a point A at (6, −6) on the curve.

Part A: What are the polar coordinates of A? Give an exact answer.

Part B: What is the polar form of the equation? What type of polar curve is this?

Part C: What is the directed distance when theta equals 2 pi over 3 question mark Give an exact answer.

Answer 1

A) In order to convert that rectangular coordinates into a polar one, we need to think of a right triangle whose hypotenuse is connecting the point to the origin.

So, we need to resort to some equations:

x ^ 2 + y ^ 2 = r ^ 2 tan(theta) = y/x theta = arctan(y/x)

Thus, we need now to plug x = - 4 and Y = 4 into that:

r= sqrt((- 4) ^ 2 + 4 ^ 2) Rightarrow r=4 sqrt 2 hat I_{s} = arctan(4/- 4) hat I , = arctan(4/- 4) + pi hat I ,= - pi/4 + pi

Note that we needed to add pi to the arctangent to adjust that point to the Quadrant.