Question

1. Algebraically transform r = -8sinθ into a Cartesian equation of a circle (in standard form).

Answer 1

algebraically eh.....

Answer 2

r = -8sin(t) is a straight line right?

Answer 3

no, it's a circle

Answer 4

I see it :)...polars still get me a little lol

Answer 5

Ha same here

Answer 6

lets do I table to see where it going ...can we do that?

t = 0,45,90,360....

Answer 7

when its at 90; or straight up, we get r = -8 so this is a circle about the y axis with a radius of 8 centerd 4 below the origin...

Answer 8

but I am prolly messing that up still....

Answer 9

nah..im right lol

Answer 10

well in my textbook theres a question similar to this, but i dont know how to get the last equation.

r=6sinθ
r²=6rsinθ
x²y²=6y
x²-6y+9+x²=9
(y-3)²+x²=3²

Answer 11

yeah, I was gettin to that :) just wanted to make sure I had an answer key to go by ;)

Answer 12

OOHHH sorry

Answer 13

our equation is going to end up as:

x^2 + (y+4)^2 = 16

r = -8sin(t)

Answer 14

r = -8sin(t) ; muliply by r

r^2 = -8rsin(t) ; convert to cartesians

x^2 + y^2 = -8y ; +8y

x^2 + y^2 +8 = 0 ; complete square for "y"

x^2 + y^2 +8y +16 = 16 ; convert to circle equation

x^2 + (y+4)^2 = 16 ; tada!!!

Answer 15

you see thats a typo ...8y*

Answer 16

At which step? and where did the 16 come from?

Answer 17

by now you should be familiar with a process called "completing the square". remember it?

Answer 18

step 4 I put in 8 instead of 8y ;)

Answer 19

Oh ok I see the typo

Answer 20

a complete square can be transformed back and forth between to ...... forms.

Answer 21

(x+4)^2 = x^2 +8x +16 right?

we can easily move back and forth between these equations since the are equal to each other...

Answer 22

but what is usually missing in order to "complete" a sqaure is this:

(x+___)^2 = x^2 +8x + ______

Answer 23

we need 2 numbers that are exactly the same, that add to get 8. what numbers are they?

Answer 24

4

Answer 25

good :) 4+4 = 8, that will satisfy that middle term; now we multiply 4*4 to get the last term.

4^2 = 16


(x+4)^2 = x^2 +8x + 16 you see where we got it now?

Answer 26

yeah

Answer 27

x^2 + 6x + ____

what would we complete the square with here?

Answer 28

(x+3)²=x²+6x+12?

Answer 29

real close....look again. 3+3 = 6 3*3 != 12

Answer 30

3*3 = ?

Answer 31

9

Answer 32

good :) we need to add 9 to "complete" the square

what happens when you add a number to one side of an equation? what do we do to the other side?

Answer 33

Add it on the other side

Answer 34

Exactly :) Do you see where I did that in the problem?

x^2 + y^2 +8 = 0 ; complete square for "y"

x^2 + y^2 +8y +16 = 16

Answer 35

i really should fix that typo lol

Answer 36

Haha no. That's where I'm stuck.

Answer 37

x^2 + [ y^2 +8y +____] = 0 ; complete square for "y"

what number do we use to "complete" the square for y?

Answer 38

16

Answer 39

its always gonna be half the middle term and then square it

Answer 40

8/2 = 4 -> 4^2 = 16 :) good

Answer 41

x^2 + [y^2 +8y +16] = 0 + 16
^^^ ^^^
we add 16 to both sides righ there right?

Answer 42

Yeah

Answer 43

then we just clean it up:

x^2 + (y+4)^2 = 16

Answer 44

which agrees with what we drew in the first place ;)

Answer 45

Where did the 8y go then?

Answer 46

remember we can move between the forms of a complete square?

(y+4)^2 = y^2 +8y +16

we just use the one for the other.... they are identical in value, they only look different in form.

Answer 47

Ohhh ok. I get it now!!

Answer 48

once we got a "complete" square, we use it to clean up the equation ;)

Answer 49

Ok, I finally understand this now! Thanks for helping me out!

Answer 50

youre welcome :)