Question

It can be shown that the polar curve r=a\sin(\theta)+b\cos(\theta), where ab\ne 0, is a circle. Find the center (in Cartesian coordinates) and radius of this circle in terms of a and b.

x-coordinate of center:

y-coordinate of center:

Radius:

Answer 1

What is ab\ne 0?

Answer 2

r=a sin (theta) + b cos (theta), ab is not = 0

Answer 3

Do you agree, that in cartesian coordinates the radius is \[\sqrt{(x-x_0)^2+(y-y_0)^2}\]Where (x0, y0) is the center?

Answer 4

yea

Answer 5

You could write it as a Cartesian equation by using cos(theta)=x/r and sin(theta)=y/r
and r^2=x^2+y^2

Answer 6

where does a and b come in?

Answer 7

i know that multiplying the equation by r is a legitimate step that can get me r^2=a rsin (theta) + b rcos (theta). Is that right?

Answer 8

Yeah sure you can do that.

Answer 9

this can give r^2= ay + bx right?

Answer 10

Ok and r^2=x^2+y^2

Answer 11

yes

Answer 12

so that will give x^2+y^2=ay+bx

Answer 13

ok and write that equation in this form:
(x-h)^2+(y-k)^2=r^2
where (h,k) is center and r is radius

Answer 14

so meaning (x-b)^2+(y-a)^2

Answer 15

?

Answer 16

hmm....I don't you completed the square right

Answer 17

I mean you didn't complete the square...

Answer 18

come again

Answer 19

\[x^2-bx+y^2-ay=0\]
you must complete the square
honestly you made some rules up or something

Answer 20

ok

Answer 21

can u help with that?

Answer 22

\[x^2+cx+(\frac{c}{2})^2=(x+\frac{c}{2})^2\]

Answer 23

why (c/2)^2 and not (cx/2)^2?

Answer 24

\[(x+\frac{c}{2})^2=(x+\frac{c}{2})(x+\frac{c}{2})=x(x+\frac{c}{2})+\frac{c}{2}(x+\frac{c}{2})\]
\[=x^2+\frac{c}{2}x+\frac{c}{2}x+(\frac{c}{2})^2=x^2+cx+(\frac{c}{2})^2\]

Answer 25

ok

Answer 26

what next?

Answer 27

\[x^2-bx+y^2-ay=0 \]

Answer 28

You need to complete the square for both the x part and the y part.

Answer 29

so we get (x+c/2)^2+(y+b/2)^2=0?

Answer 30

then what next? equate each part to 0?

Answer 31

hmmm....
So you have x^2-bx+b/2=(x-b/2)^2

Answer 32

But whatever you add to one side you add to the other .

Answer 33

x=b/2,y =a/2 and r=sqrt(a^2+b^2)/2

Answer 34

YEP that is right.

Answer 35

thank you