Find the points of intersection of the graphs of the equations.r = 9(1 + sin(𝜃))r = 9(1 − sin(𝜃))r ≥ 0, 0 ≤ 𝜃 < 2𝜋
Question
Find the points of intersection of the graphs of the equations.r = 9(1 + sin(𝜃))r = 9(1 − sin(𝜃))r ≥ 0, 0 ≤ 𝜃 < 2𝜋
Solution
To find the points of intersection of the two polar equations, we need to set them equal to each other and solve for θ.
So, we have:
9(1 + sin(θ)) = 9(1 - sin(θ))
Solving this equation gives us:
1 + sin(θ) = 1 - sin(θ)
2sin(θ) = 0
sin(θ) = 0
The solutions to this equation in the interval 0 ≤ θ < 2π are θ = 0 and θ = π.
Now, we substitute these values into the original equations to find the corresponding r values.
For θ = 0:
r = 9(1 + sin(0)) = 9(1 + 0) = 9
r = 9(1 - sin(0)) = 9(1 - 0) = 9
So, one point of intersection is (9, 0).
For θ = π:
r = 9(1 + sin(π)) = 9(1 + 0) = 9
r = 9(1 - sin(π)) = 9(1 - 0) = 9
So, the other point of intersection is (9, π).
Therefore, the points of intersection of the graphs of the equations are (9, 0) and (9, π).
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