00:01
In this problem, we are given a couple of polar equations.
00:05
We are going to sketch the graph of each of these equations and we are also going to convert them into rectangular coordinates.
00:16
We have r equal to 2 sine theta.
00:23
Let us use the given information.
00:26
Let me just write down over here.
00:28
X is equal to r cosine theta.
00:32
Y is equal to r sine theta.
00:37
R squared is x squared plus y squared.
00:41
And theta is equal to r tangent y over x.
00:49
Now we have r equal to 2 y over r.
00:54
So we have 2 y equal to r squared equal to x squared plus y squared.
01:06
X squared plus y squared minus 2 y equal to 0.
01:11
Then i want to rewrite this equation as x squared plus y squared minus 2 y plus 1 equal to 1.
01:22
So x squared plus y minus 1 squared equal to 1.
01:28
This is a circle of radius 1 centered at x equal to 0 and y equal to 1.
01:34
So let's sketch the graph roughly.
01:39
X here, y here, 1, 2, 1, minus 1.
01:48
Something like this.
01:53
Next, we have r equal to minus sine theta.
02:02
Now we have minus y over r.
02:07
So minus y is equal to r squared equal to x squared plus y squared.
02:16
Then we have x squared plus y squared plus y equal to 0.
02:21
Let me rewrite this equation in the following form.
02:25
X squared plus y squared plus 2 times 1 half y plus 1 half squared equal to 1 half squared.
02:38
So we have x squared plus y plus 1 half squared equal to 1 half squared.
02:48
This is a circle centered at x equal to 0, y equal to minus 1 half and it has radius 1 half.
02:59
So we have something like x here, y here, minus 1 half, minus 1, 1 half, minus 1 half, 1 half, minus 1 half.
03:19
Something like this.
03:21
Next, we have r cosine theta equal to minus 4.
03:32
So the left hand side is just x.
03:35
So we have x equal to minus 4.
03:39
And the sketch is x here, y here.
03:46
It's a vertical line passing through the point x equal to minus 4.
03:52
Next, we have r equal to 3 cosine theta.
04:00
We have 3x over r.
04:03
So 3x is equal to r squared equal to x squared plus y squared.
04:10
So we have x squared minus 3x plus y squared equal to 0...