Determine the θ-values for the points of intersection of the graphs of the polar curves r = 8 cos (θ) -2 and r = 6 cos (θ) over the interval [0, 2π).
Enter an exact answer and separate multiple answers with commas, if necessary. If the origin is a point of intersection, do not include it in your answer.
Answer (2)
Set the polar equations equal: 8 cos ( θ ) − 2 = 6 cos ( θ ) .
Solve for cos ( θ ) : cos ( θ ) = 1 .
Find θ in [ 0 , 2 π ) : θ = 0 .
The intersection point is at 0 .
Explanation
Problem Setup We are given two polar curves r = 8 cos ( t h e t a ) − 2 and r = 6 cos ( t h e t a ) , and we want to find the values of t h e t a in the interval [ 0 , 2 π ) where they intersect, excluding the origin.
Equating the Polar Equations To find the intersection points, we set the two equations equal to each other:
8 cos ( t h e t a ) − 2 = 6 cos ( t h e t a )
Solving for cos ( t h e t a ) Now, we solve for cos ( t h e t a ) :
2 cos ( t h e t a ) = 2
cos ( t h e t a ) = 1
Finding the Values of t h e t a We need to find the values of t h e t a in the interval [ 0 , 2 π ) that satisfy cos ( t h e t a ) = 1 . The only value that satisfies this condition is t h e t a = 0 .
Checking the Origin We are asked to exclude the origin as a point of intersection. Let's check if either curve passes through the origin. For the first curve, r = 8 cos ( t h e t a ) − 2 = 0 implies cos ( t h e t a ) = f r a c 1 4 . For the second curve, r = 6 cos ( t h e t a ) = 0 implies cos ( t h e t a ) = 0 . Since the curves do not simultaneously pass through the origin at the same t h e t a value, we don't need to exclude the origin.
Final Answer Therefore, the only value of t h e t a where the two curves intersect is t h e t a = 0 .
Examples
Understanding intersections of polar curves is crucial in fields like radar technology, where identifying overlapping signals from different sources requires precise mathematical analysis. For instance, in air traffic control, radar systems use polar coordinates to track aircraft positions. Determining the points where radar signals intersect helps to differentiate between multiple aircraft and avoid potential collisions, ensuring safer air travel.
The curves r = 8 cos ( θ ) − 2 and r = 6 cos ( θ ) intersect at θ = 0 , but this point is the origin, which is to be excluded. Thus, there are no valid intersection points to report for the interval [ 0 , 2 π ) .
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