Find the point(s) of intersection for the polar curves with equations r = 6 cos θ and r = 4 − 2 cos θ.

a) π/6,-π/6
b) π/3,-π/3
c) π/2,-π/2
d) 0,π

I did:

6cosθ= 4 - 2cosθ
For the circle will be 6cosθ is:
r=6cosθ
r^2=6rcosθ
x^2+y^2=6x
but I NEED for the cardioid to get my 2 equations and then I will use my first equation to replace in the second equation with 6x

To find the point(s) of intersection between two polar curves, we need to find the values of θ that satisfy both equations.

The given polar curves are:

1. r = 6 cos θ
2. r = 4 - 2 cos θ

Setting the two equations equal to each other, we get:

6 cos θ = 4 - 2 cos θ

Adding 2 cos θ to both sides:

8 cos θ = 4

Dividing by 8:

cos θ = 1/2

Now, we know that cos θ = 1/2 when θ is π/3 or -π/3 (using the unit circle). So, the possible values for θ are π/3 and -π/3.

To find the points of intersection, we substitute these values of θ back into one of the original equations. Let's use r = 6 cos θ.

When θ = π/3:
r = 6 cos (π/3) = 6 * (1/2) = 3

So, we have one point of intersection: (3, π/3)

When θ = -π/3:
r = 6 cos (-π/3) = 6 * (1/2) = 3

So, we have another point of intersection: (3, -π/3)

Therefore, the correct answer is:

a) π/6, -π/6

To find the points of intersection for the given polar curves, we need to set the two equations equal to each other and solve for θ.

First, let's set the two equations equal to each other:
6 cos θ = 4 - 2 cos θ

Add 2 cos θ to both sides:
8 cos θ = 4

Divide both sides by 8:
cos θ = 1/2

To find the values of θ, we need to examine the unit circle or use a calculator. The angles that have a cosine value of 1/2 are π/3 and -π/3.

So, the possible values for θ are π/3 and -π/3.

Therefore, the point(s) of intersection for the polar curves are (π/3, r) and (-π/3, r).

To find the values of r at these angles, substitute the values into either of the original equations. Let's use the first equation r = 6 cos θ:

For θ = π/3:
r = 6 cos(π/3)
r = 6 * (1/2)
r = 3

For θ = -π/3:
r = 6 cos(-π/3)
r = 6 * (1/2)
r = 3

So, the point(s) of intersection for the polar curves are (π/3, 3) and (-π/3, 3).

Therefore, the correct answer is:
b) π/3, -π/3

huh? This is just Algebra I. You want to solve

6cosθ = 4 - 2cosθ
8cosθ = 4
cosθ = 1/2

Now you're home free!